Baes on this general description, the firearm in question could be any of the standard rifles in use by the army. The Austrian Lorenz (cal. 54) would be one choice. The others would be the the Enfield (cal. 577) and Springfield (cal. 58).
The three firearms named above were standard weapons used throughout the war by Confederate mounted troops (and some on the Federal side). As noted earlier, they were totally unfit for cavalry service.