Harold Wentworth, American Dialect Dictionary (1944) confirms your surmise that the word chance in this context means "(large) number":
chance, n. ... 2. A quantity or number, usu. in 'a right smart chance,' meaning an indefinitely large one; also, a crop or herd, distance company;—used also absolutely, & with monosyl. adjs. (smart, right, fine, nice, great, good, whole).
1823 West 1827–37 smart chance. Provincial. Sherwood Gaz. of Ga. 1829–30 s.U.S. 'He lost a right smart chance of blood.' Vulgarism. Dunglison Glossary. 1856 I'm acquainted with a right smart chance of gals in Keokuk. Twain. Before 1883 Va. he expects to raise 'a fine chance of curcumbers.' Bagby Old Va. Gent. 1895 s.e.Ky., e.Tenn., w.N.C. There's a right chance o' snack houses down to Bakervul. 1899 s.e.Va. Warwick Co. He's got a right smart chance of children. Green. 1902 s.Ill. right smart chance of taters or money. 1903 s.e.Mo. There will be a smart chance of peaches this season. 1908 e.Ala., w.Ga. 'He's got a right smart chanct of cotton.' Common in 'right smart chance,' but often in other collocations. 'We caught a nice chance of fish.' 1915 s.w.Va. Scott Co. We made a great chance of apple butter this year. 1925 w.cent. W.Va. 'That is a right smart chance of cattle." ...
The narrator in the song "I'm a Good Ol' Rebel" says that he fought for General Lee, which means that he was enlisted in the Army of Virginia during the U.S. Civil War—and which puts him geographically in the midst of the upper South at the right time to be using chance in the specified way. The only thing not consistent with Wentworth's coverage of chance is that the word appears in the song lyrics without a modifier such as right, smart, or nice. But this usage may be as much a response to requirements of conciseness in song lyrics as anything else.