What you're looking for is the Union "Order of Battle" for the 20th corps in the Atlanta Campaign.
Here ya go, I've already done your heavy lifting. Scroll down the page and it will list the cavalry attached to the 20th corp. There were three divisions of Cavalry.
You can already discount the 7th Illinois Cavalry, they didn't take part in the Atlanta campaign. They were running around in TN & MS--my gg grandfather was with them. And not all of companies of the 7th had Smith's, they also had Sharps carbines.
en.wikipedia.org
If you are sure that the camp site was of the 20th Corps / 3rd division, then you have 3 possibilities:
the 10th Ohio, the 3rd Ky, and 4 companies of the 3rd Indiana cavalry, but check it for yourself to make sure I didn't miss something.