IANaLb...
It all depends on what your contract says and what rights you signed away.
I think I've seen bands record a song multiple times on multiple labels, but I suspect that's going from a smaller label to a larger (where the larger label has a lot more clout to dictate terms).
I would assume that there's no way that you're going to be able to release the same recording on a new label unless the new label buys the rights from the old entirely.
Again, it depends on what your contract says. If you think there's a remote chance you'll get sued, contact a REAL lawyer.