The Yiddish ballad Papirosn consists of lyrics set to a pre-existing tune.
I think the same may be true for Ale Brider because I've heard a different set of words sung to the same tune (Bay Mayn Rebbe Iz Gevezen a Ganeyva, or something like that). I've also heard a song, Liebe Tochter, to the tune of Tanz, Tanz, Yiddelekh, and I'm pretty sure the tune came first.