Did he have a middle name? There are lots of John [Middle Initial] KIDD's in the war, including some from AL, as well as a few [Initials Only] KIDD's (but no plain "Johns" listed):

Kidd, John B., AL 3rd Reserves, Co. "C"
Kidd, John B., AL Cavalry Hardie's Bn. Reserves, Co."A"
Kidd, John G., AL 13th Infantry, Co. "E", a Sgt.
Kidd, John M., AL 3rd Reserves, Co. "B", a 2nd Lt.
Kidd, J., AL 3rd Infantry, Co. "B"
Kidd, J.N., AL Militia, 2nd Regiment, Co. "A"
Kidd, J.N., AL Mobile City Troops