Ron, I'm not at home so I don't have access to my normal resources. By the end of the War It is most likely they had a mix of three weapons. Enfield, one version or another of the 1861 Springfield or the Austrain Lorenz. Early War (pre 1863) percussion conversion of the 1816 Springfield, 1842 Springfield or if they were particularly lucky M1841 (Mississippi Rifle).
When I get home I'll take a look see and see if I have any refernce to what they might have carried. The list is fairly complete for any
CS unit w/ the ANV but most other units are a guess at best. AoT had a large proprtion of Austrian Lorenz's. Best guess 1/2 to 2/3 were armed w/ them. The rest was a mix of Enfields and Springfields... though 61 Springfields were relatively uncommon. The
CSA Army in the West was kind of the red headed step child of the
CSA w/ the ANV getting most of the most modern arms. The M1842 was M1816 percusion conversions were very common in Alabama Arsenals prior to the war. There were also a good many M1841's in state Arsenals. Though most of those went to the first Alabama Regiments raised... I suspect most of those had been issued by the time the 31st was raised. Inital guess would have been M1842, M1816 conversions or a mix therof. As better weapons became available they were issued.
You might check the regimental history as they often contain references to the weapons issued, especially if they were considered poor quality.
Hope that helps some.