The 1860 Federal Census for Winston County shows one C.W. Miles, son of E.M. Miles. This young man was listed as being born about 1844.

The Works Progress Administration source manuscripts for Winston County show one C. Munroe Miles as being a member of Company E, 5th Mississippi Infantry. See:
http://www.datasync.com/~jtaylor/5msie.htm

The penmanship on the census isn't the best and I'd suspect these are the same person. That's not to say that this is Calvin Miles but it is a possibility. A little more genealogy research will help, as you've suggested.