Well, keep posting, with pictures, not guesses as to model, and you will learn a lot about them here.
That is really a neat little gun, even though rebored and reblued with stock repair.
It would be nice if you had a good gunsmith to check it out. There are only a few things to go wrong with a Winchester Single Shot but some of them show up when converted to a modern caliber.
Lacking professional evaluation, I would work up through the scale of ammo power.
If it shot .38 Specials ok, I would try .38 +P. If they showed no sign of primer cratering, primer backing out, or casehead separation, then on to a Magnum.