The amounts of agent orange, in units of 55-gallon (208 liter) drums by company is as follows:
- Dow 78,235
- Monsanto 67,065
- Hercules 49,945
- Thompson-Hayward Chemical (T.H. Agriculture and Nutrition ) 21,055
- Diamond Alkali /Shamrock 12,555
- US Rubber (Uniroyal) 11,635
- Thompson Chemicals 7,185
- Agrisect 1,875
- Hoffman-Taff 410
The above data is from table 2.2 (page 44) of The History, Use, Disposition and Environmental Fate of Agent Orange by A. L Young (2009).
The book also repeats this data in table 5.6 (page 173), breaking it down by fiscal year. For the first two years, 1963 and 1964, Monsanto was the only supplier. The statement in the OP "When Dow couldn't manufacture it fast enough, the government turned to overflow suppliers" is thusly untrue. In 1965, Dow and Hercules joined Monsanto in the production effort. The other companies started still latter.
While Dow produced more Agent Orange, it produced agent orange with a lower concentration of dioxin contamination. The same book provides data in table 5.11 (page 183), for each of the top 5 producers, broken down by year.
- At Monsanto, dioxin concentration ranged from 1.04-11.53 ppm
- Diamond Alkali/Shamrock 0.53-8.27 ppm
- Dow: undetectable to 0.63ppm
- Thomas-Hayward: 0.32ppm
- Hercules: undetectable to 0.03ppm (undetectable in 47 of 48 samples)
So there was vast variance in the concentration of the toxic substance, dioxin.
In 1984, several companies that manufactured agent orange reached a settlement with veterans who had sued for damages due to agent orange exposure.
The companies involved and percent payment was as follows:
- Monsanto: 45.5%
- Dow: 19.5%
- Diamond Shamrock: 12%
- Hercules 10%
- T.H. Agriculture and Nutrition 6%
- Uniroyal 5%
- Thompson Chemical 2%
The percentages being "according to a formula based on the amount of Agent Orange each firm produced and how much the substance was contaminated with dioxin".
The above data and quote is from Spartanburg Herald-Journal 13 May 1984 page A5.
In conclusion, Dow produced more agent orange than any other company, Monsanto being a close second. Monsanto settled for the most liability, based upon a formula that also considered how contaminated each company's agent orange was with dioxin.