Hugh, the "real" 1st Arkansas Regiment never served in the Trans-Mississippi Department. Therefore, the 1st Arkansas to which Odom was attached could have been either the 37th Arkansas (aka 1st Trans-Mississippi/1st Arkansas) or any one of a half-dozen mounted regiments that styled themselves as the "1st Arkansas Cavalry." None of these outfits have any surviving records from the period that Odom would have been attached.

Another possibility is the 1st Consolidated Arkansas Infantry (TMD), which has no surviving records whatsoever. My gut feeling is that this is probably the "1st Arkansas" referred to. It was comprised of exchanged Vicksburg and Port Hudson prisoners, and was commanded by Col. Jordan E. Cravens. However, with no records to work from there's just no way of knowing for sure.