Most of my ancestors served in the Confederate army in units from either Mississippi or Alabama. However, one man, a resident of Lawrence County, Alabma, chose to serve with the 1st Alabama Cavalry, US Volunteers. It is not easy to track all the members of this unit as many of them used an alias not just to protect themselves, but also to protect their families back home from retribution should they be captured. My guess is that the majority of men hiding their identity during the war did so for that reason. Still, having once thought that it might be nice to wipe the slate clean and start over, I'm sure that the war gave many men the opportunity to do just that - disappear from family, debts, or any other sort of disagreeable situation, in order to assume a new identity and move on somewhere else after the war.
Below is a link to a forum for discussion of Southern Unionists. Posting your question on that forum might get you some specific examples.