There cannot be any direct evidence, as far as we know. And the authors of the billboard know surely less than that.
But the circumstantial evidence reveals some interesting tidbits. These make the claim that a Judean or Galilean women around the year 1 wore a hijab or any kind of veil highly unlikely:
Judean women were clearly interested in the arts of adornment, as is evident from the cosmetic equipment, including combs and ring keys, found at Masada. Hair found at the same site indicates that women wore their hair long, perhaps in a single braid, but no doubt wealthier women were influenced by the fashions of the day, arranging their hair in more ornate ways (Edwards: 237). It is only later in the rabbinic writings that women are exhorted to cover their heads (Ketub. 72a) out of general concerns for modesty and to indicate a transition from girlhood to womanhood (Bronner: 466). The Talmud claims that a woman going out with her head uncovered is in violation of biblical law. This is because of Numbers 5:18, which describes a ritual stipulated for a suspected adulteress. The priest must either unbind or uncover her hair. The Talmudic authors argue that he was uncovering it, therefore presuming that married women normally went about with their head covered. However, other interpreters debated whether or not the specific word in question, parah, meant to loosen or to uncover. Regardless, hair covering for married women eventually became a widespread practice in various forms of Judaism. Midrash adds the attractiveness of Eve to the reasons why she ate the fruit and apparently seduced the man (Gen. Rab. 17:8; see Bronner: 470–71), which provides further justification for women to cover their heads so that they will not tempt men who cannot resist them.
Alicia J. Batten: "Clothing and Adornment", BiBlical Theology BulleTin Volume 40 Number 3 Pages 148-159. DOI: 10.1177/0146107910375547
So, we have first a much more fashionable uncovered head, albeit presumably just when not working the fields in rural settings.
Then we have a later development within the emerging Judaism that slowly went about to more or less enforce this habit.
And this is just for married women. As the story goes, when Mary became the mother of Jesus, she was not married.
(Depends a bit on translation, of the *very few passages we have from the bible for that Mary at all. Luke (OJB): "To a betulah (virgin) given in erusin (betrothal, engagement) to an ish from the Beis Dovid named Yosef [ben Dovid], and the shem of the almah was Miryam." But the original is Greek (SBLGNT): πρὸς παρθένον [a]ἐμνηστευμένην ἀνδρὶ ᾧ ὄνομα Ἰωσὴφ ἐξ οἴκου Δαυὶδ, καὶ τὸ ὄνομα τῆς παρθένου Μαριάμ. And neither means "married". As the two previous paragraphs should make clear, if a later development prescribed head coverings for married women, the scant evidence from the bible is apparently not much in favour of that Mary wearing a head cover.
As Mary would have lived 700 years before Islam was invented, we can safely conclude that on the one hand fashion styles varied, and the early hijab is not the one of today, nor the only form of headdress allowed or prescribed in Islam anyway. Did Mary wear a burqa? Afghan taliban billboards would claim that, for sure, at least for as long as billboards are allowed under their rule.
That a Mary would have followed this prescription, or honored it, is not unlikely. As it is the central theological argument for a hijab, in some interpretations, but does not talk about head or hair:
Sura 24,31] And say to the believing women that they should lower their gaze and guard their modesty; that they should not display their beauty and ornaments except what (must ordinarily) appear thereof; that they should draw their veils over their bosoms and not display their beauty except to their husbands, their fathers, their husband's fathers, their sons, their husbands' sons, their brothers or their brothers' sons, or their sisters' sons, or their women, or the slaves whom their right hands possess, or male servants free of physical needs, or small children who have no sense of the shame of sex; and that they should not strike their feet in order to draw attention to their hidden ornaments. And O ye Believers! turn ye all together towards Allah, that ye may attain Bliss.
The actual history of the hijab is much more filled with circumstance than never changing doctrine:
The world of clothing in Islam is subjected to both normative and juristic restrictions, creating tension between juristic demands, local realities, and human wishes. Some of these main juristic instructions are the need to cover the pubis, to keep the laws of modesty, to keep the patriarchal structure, and not to act with arrogance but at the same time demonstrate God’s greatness through clothing. Muslim jurisprudence thus should be interpreted as an outcome of this dialog, while other texts such as adab literature and poetry, in addition to physical remains, present a rich world of personal appearance, sometimes in contradiction with juristic norms and restrictions.
Islam was created and developed in the Arabian Peninsula and was first aimed at its pagan Arab inhabitants who became the believers. Through Jahiliyya (pre-Islamic) times, most of those Arabs were nomads, with the exception of some permanent settlements such as Mecca. The major variables that influenced the code of clothing were climate, life needs, social codes, religious demands, and socioeconomic manifestations. One of the most frequent motifs of Jahiliyya-Arabic poetry is the praise of personal appearance, especially of women. The first written and organized code of proper Muslim clothing is contained in the holy scripture of the Qurpan, which devotes several verses to clothing, adornment, and modesty. According to Qurpanic verses, clothing and adornment are God’s grace to his believers, aimed at covering their pubis and adorning themselves (16:81, 7:26). The other use of clothing in the Qurpan is for methodological and didactic purposes, as clothing is a part of the reward in heaven or alternatively a part of the punishment in hell (18:31, 35:33, 44:53, 76:21, 22:23).
The great conquests of Islam and the spreading out of the religion beyond the Arabian Peninsula brought under its rule different cultures and ethnic origins, with their own worlds of clothing. Hadith collections, Quranic exegesis, and fiqh compendiums were aimed at providing guidance to Muslim communities in the post-Quranic age, all reflecting extensions and changes of the Qurpanic verses based on time, place, and needs. This means that the Quranic Muslim code of clothing was extended and reinterpreted, thus reflecting the tension between adornment and self-expression, arrogance and modesty, norms and reality. The results of these intersections was a mixture of Arab clothing and Turkish and Iranian clothing, based on geography, ethnic origin, and mutual influences.
Hadas Hirsch: "Clothing and adornment" p 221, in: Richard C. Martin et al (Eds): "Encyclopedia of Islam and the Muslim World", Gale: Farmington Hills, 2016.
For example, the hijab verse (33:59) is reportedly occasioned by the suggestion by Umar ibn al-Khattab, portrayed in hadith literature as a fierce advocate of female subjugation, that the wives of the Prophet be veiled and confined. Gender dynamics within the community as reflected in gender-relevant asbab al-nuzul (occasions of revelation) literature remains to be fully analyzed for its implications for feminist hermeneutics of the Quran.
(Ghazala Anwar: "Feminism" ibd, p373.)
Even stranger in relation to the billboard is the story brought down via Anas ibn Malik, a supposed companion of Mohammed, who reported that when the prophet married yet another wife called Zaynab bint Jahsh those suras came down to earth. Furthermore, the other so-called veil verse (Ayat al-Hidschab; see 33:53), which supposedly obliged the wives of the Prophet to wear the veil, is said to have been revealed on the wedding night with Zaynab. Conveniently just after Mohammed got jealous as the many guests laid eyes on the bride. Some traditions also connect 66:1 with the jealousy of the rest of the Prophet's wives toward Zaynab.
(C.E. Bosworth, “Zaynab bt. K̲h̲uzayma”, in: Encyclopaedia of Islam, Second Edition, Edited by: P. Bearman, Th. Bianquis, C.E. Bosworth, E. van Donzel, W.P. Heinrichs. Consulted online on 23 November 2018 http://dx.doi.org/10.1163/1573-3912_islam_SIM_8150)
Another story holds that Mohammeds wives were frequently approached at night by lusting strangers on the streets. After being caught they justfied their unruly behaviour with the fact that they mistook the prophet's wives for slaves, since they were unveiled.
Also confer to Claudia Knieps: "Geschichte der Verschleierung der Frau im Islam", Ergon: Würzburg, 21999. (Pages 162–175, 190–200.) and Susanne Enderwitz: Der Schleier im Islam. In: Feministische Studien. Vol 2, 1983, p. 96 (DOI).
Fatima Mernissi: "Women and Islam: An Historical and Theological Enquiry", Blackwell: Oxford, 1991, p 85–101: "The hijab, the veil".
If we look again at the facts scattered throughout this chapter, it comes down to saying that the Prophet, during a troubled period at the beginning of Islam, pronounced a verse that was so exceptional and for the Muslim religion that it introduced a breach in space that can be understood to be a separation of the public from the private, or indeed the profane from the sacred, but which was to turn into a segregation of the sexes. The veil that descended from Heaven was going to cover up women. separate them from men, from the Prophet. and so from God. Having clarified this aspect of the subject - the linguistic, social, historical, and religious reality of the hijab - should we not ask ourselves how the Prophet, who felt such an absolute and radical need to protect his privacy, lived?
Meaning that previously not even the wives of Mohammed honoured the hijab?
The full treatment for this development o head dresses in Judea and surroundings is nicely condensed in:
Cynthia Baker: "Rebuilding the House of Israel: Architectures of Gender in Jewish Antiquity", Stanford University Press, 2002.
Read as written, the words convey only the following demands: that, when going out, a wife must cover her head (early modern Euro-American practice dictated similar protocols for men and boys, although without dire threats of legal repercussion), and she must take care that her body is covered by her clothing; that she must not go out to spin yarn in the shuk (the Mishnah adds that she must not engage in [intimate] conversations with other men); and that she must take care while at the common bathhouse to bathe and wash in such a way that she is not doing so in the presence of “everyone”— men and women alike. While it is not appropriate to read such rabbinic prescriptions either as simple descriptions of common practice or as benevolent concern over women’s “modesty,” neither is it appropriate to exaggerate them into a cultural ideal demanding the confinement of women or their banishment from public places — in fact, the prescriptions in these passages are incomprehensible unless both women and men are understood to occupy all of the spaces mentioned, together, and at the same times.
Although the actions and codes described in this passage might or might not reflect widespread social custom (we have few other forms of evidence by which to judge), it is likely that the images conveyed are at least consistent with rabbinic constructions of customary practice. To the extent that this is the case, we might hypothesize that the married woman’s distinguishing mark would likely be her headgear.89 In fact, “a hat for her head” is among the clothing items that a husband is enjoined, in the Mishnah, to provide for his wife,90and if he does not, the Tosefta asserts that others may buy goods from her (even if the profits rightfully belong to her husband) in an amount that would enable her to obtain one. Moreover, wives who “go out with head uncovered” are threatened with divorce and financial loss; their husbands, as we saw above, are reviled as “evil”; and heavy fines are decreed against anyone who would dare remove a woman’s headgear against her will.
Detailing that this is indeed just another incidence of one sect copying concepts of social control from another. Only in this case that concept as detailed developed too late for a Mary in question and the copyists are blissfully unaware of it.
The currently oldest depiction of Mary is thought to be from Dura-Europos, today in Syria:
This image of a woman drawing water from a well once decorated the baptistry of an early Christian house-church in Dura-Europos. It could be the earliest known image of the Virgin Mary, according to a scholar from Fordham University. The image has been rendered in black and white on the right to better show the outline of the figure.
Yale Art Gallery painting might be oldest known image of the Virgin Mary, 2016
In that image at least, headdress seem to be absent. Other old images, either much younger or much farther away geographically, from the Sea of Galilee or Jerusalem indicate a general trend: that Mary was depicted with headdress only later, when the customs of the painters changed.
XV14 - Roma, Museo civiltà romana - Adorazione dei Magi - sec III dC - Foto Giovanni Dall'Orto