In addition to being a "regular soldier" he was a saddler. Look at it from that point of view. When he wasn't drilling or standing guard, he was maintaining the saddles and riding gear for the company.

U.S. Veterans Gravesites, ca.1775-2006
Name: John Jacob Bittmann
Death Date: 13 Oct 1864
Cemetery: Andersonville National Historical Site
Cemetery Address: Route 1, Box 800 Andersonville, GA 31711-0000
Buried At: Section H Site 10835

NPS shows:

John I. Bittmann
Regiment Name 1st Michigan Cavalry
Side Union
Company C
Soldier's Rank_In Pvt.
Soldier's Rank_Out Pvt.
Alternate Name John J./Bettman
Film Number M545 roll 4