I would look at where each of the possible regiments were formed and check for Smith Co. You should also look for what the regiments were reorganized as or consolidated into.

12th Regiment, Tennessee Cavalry (Green's): 12th Cavalry Regiment [also called 1st Partisan Rangers] was organized behind Federal lines in February, 1863. The men were from the counties of Fayette, Tipton, Shelby, Haywood, and Gibson. Some of its members became part of the 3rd (Forrest's Old) Tennessee Cavalry.

31st (A.H. Bradford's) Infantry Regiment was assembled in October, 1861, at Camp Trenton, Gibson Couty, Tennessee. Its members were recruited in the counties of Weakley, McNairy, Haywood, Gibson, Madison, and Decatur.

22nd (Barteau's) Cavalry Regiment, originally and often called 2nd Regiment, was organized in June, 1862, by consolidating the 1st and 7th Tennessee Cavalry Regiments.

7th Cavalry Battalion was organized at Camp Jim Davis, Macon County, Tennessee, in November, 1861, with six companies. Its members were from Sumner and Smith counties.

I would then check the 1860 census for other A.J. Mintons and see where they were living. You then might be able to rule out one or the other. Unfortunately, Gibson county overlaps both of your potentials so if you find an A.J. in that county, that won't help matters.

You can also check both of the A.J.'s records as sometimes residence is given on oath or other records. If your A.J. is the only one living in Smith Co and that happens to be listed, he might be your guy.

Sorry for all this rambling but hope some of this helps! Good luck!