Was he living in Marshall County? There were a Moore family living there. The Director of the Marshall County Museum in Holly Springs, Jim Moore, is a descendant of one of the veterans. Two of the MOORE family served in the 17th Mississippi Regiment and one wrote a journal that has been published---a great read.
You need to try to find where he was living in 1860 Census. This will tell us his Age which we might get lucky and find in a record. It will also help direct us to some regiments. Right now, we don't know if we should start with Alabama or Mississippi Regiments.
James R. Moore enlisted as the captain of Company D, 5th MS Infantry. He was promoted to major, date not known, so he is found under Field & Staff upon receiving that elevation.