Your ancestor's name is spelled A.J. Gardener, Co. E, on the roll I find for Demoss' 10th Tn. Cavalry in the Soldiers and Sailors system, and this also indicates he had an alternate name (by error probably) as J.A. Gardner. Both these names show in the roll of Company E and appear to be the same person. Finding the records of J. A. Gardner might reveal more about your ancestor.

Use this URL and you can view all soldiers in this regiment.
http://www.itd.nps.gov/cwss/Soldier_Results.cfm?maxcount=1283&start=1&unitcode=CTN0010RC01

Hope this has been helpful.