For what it's worth, the late Larry Koller, one of the best known gun writers in the1960's, tested Colt's then-new Diamondback .38 on a deer. He used a 125 grain bullet, but I can't recall the type. It was almost surely a handload back then. The gun had a four-inch barrel.
He killed a whitetail with it, and wrote that it did fine. He lived in New York, so it wouldn't have been a small one, especially.
For Coues deer, (if you get close enough), Key deer, tropical brocket deer, small Texas Hill Country bucks, etc., the .357 should be plenty. For northern deer or the big south Texas animals, a .357 needs to be just right for the circumstances, and I wouldn't push the range.
A friend of mine began deer hunting with a Winchester M92 in .32/20. He ambushed deer in southern Arkansas and shots were usually within 40 yards. He said it killed well. If a deer is big, the range is long, or it's adrenalized and moving, the .357 is probably marginal.
Lone Star