Drafting is critical, early and often. The main advantages the Union has are numbers, transportation and supply. You must exploit all three. FYI I was a beta play tester. Although I don't claim to be especially good at this game, I do have lots of experience in a dozen PBEM against 5 different opponents, mainly from the Union side.
In my current game against a very experienced, excellent player I drafted with the following results:
July 1861 53 militia, August 32
January 1861 36 militia, February 24
August 1862 (I think I was hoping for emancipation so I delayed the draft to save the PPs) 32 militia September 28
February 1863 17 militia, March 8
The value of the draft is that you gets significant numbers of reinforcements at one time rather than just dribbling in as bits and pieces. You can also set up training centers for these masses of people. I try to train in the north rather than the south as my prejudice is that it takes less transport to get them there and they suffer less attrition. If you need Kentucky garrisons, that is also a good place to train them in great numbers. Often these newly trained regulars can save transportation by simply "bumping" forward one region, displacing another regular forward until an unspotted regular arrives at the front.
USCT (United States Colored Troops): I commenced raising them in March 1863. At s population cost of 6 they are always relatively cheap. By capturing southron regions, you are sometimes able to raise a garrison for them from the black recruiting you do in that region; you "make war pay for war" as it were.
My USCT troops were March '63 14, April 10, May 9, June 4, July 6, August 4, Sept 6, Oct 4, Nov 9, Dec 1. My garrisons for Dec 1863 were about 100 units, so these 53 USCT covered almost half of that. Besides, I enjoy the irony.
USCT recruiting tends to reflect the progress of your offensive; the black recruits accrete in their cities, when you capture those cities you will typically get 1 USCT in each of the next 2 building phases. Very few southern regions require more than 3 to 4 garrison, so you see, you are raising your own garrison.
"L'audace, l'audace, toujours l'audace."