Henry Belden's Songs and Ballads Collected by Missouri Folk-Lore Society has some Civil War songs collected in the early 20th century. There is some fairly obscure material, including some Yankee songs parodied with pro-Confederate lyrics.

I think the best Civil War songs are the ones that are not about the war, but about what soldiers on both sides were homesick for ("Girl I Left Behind", "Lorena", etc.