According to the Civil War Soldiers and Sailors System
http://www.itd.nps.gov/cwss/soldiers.htm
There was a Joseph L Markey who was in Co E, 24th MO Infantry that was in the Chattanooga area at that time. Also there was another (or the same) Joseph L Markey in Co K, 26th MO Infantry that was also there at the same time. These could be the same person.

There was also a Joseph M Welch in Co G, 7th Battalion Mississppi Infantry, and a James M Welch in Co A, 1st (Colqitt's)Arkansas Infantry. Both were involved in the Chattanooga/Atlanta Campaign.