While you wait for more answer, this thread might be of help to you. I think the answer from /u/lordtiandao is quite insightful.
It has usually been assumed that the massive figures in Sima Qian were “notional” as Lewis puts it. Most of the huge figures Lewis cites come from the Zhanguo ce, which is more a guide to rhetorical persuasion than a technical military manual, and the one million figure for Qin seems to clearly mean “the largest possible army”.
In the early Warring States both the Sunzi and the Mozi suggest 100,000 as the number of a typical field army. At Maling in 341 BCE Wei was defeated with an army of about 100,000.
At the very end of the period army size and the length of campaigns both increased, although even then Lewis suggests that casualty figures of 20,000 to 40,000 were more common.
Part of the problem with these figures is that by late Warring States battles were no longer affairs of day or two. The Changping campaign lasted 3 years and stretched over a front of a hundreds of li. The war between Qin/Wei/Hann and Chu/Qi involved offensives on four fronts over almost the whole Yellow River plain. The siege of Handan lasted 3 years, and many of these campaigns involved massive fortifications. If you consider the total number of men mobilized to fight in and support these armies…Well, you probably still don’t get a million, but you do get a big number, especially if you are thinking of the number of men mobilized for the entire war, rather than those present on a single day of battle. Lewis is fairly typical in not trying to claim he knows the correct numbers, but to point out that all sources seem to show that armies were large early in Warring States and even larger by the end.
Sources
Mark Edward Lewis “Warring States Political History” in Loewe, Michael., and Edward L. Shaughnessy. The Cambridge History of Ancient China : From the Origins of Civilization to 221 B.C. Cambridge: Cambridge University Press, 1999.