Might consider a .40

You could probably deer hunt with the .357 you've got, not that you ever will, but I think you need a revolver with a 3" or 4" barrel, with a minimum caliber/diameter, as well maybe a minimum energy level, think the .357 qualifies though, only thing would be if that barrel was long enough.

I think a .40 would be my choice for a cartridge with no other circumstances in a semi-auto.

If you have a wife who wants to shoot it, or a kid, or it's a compact model, etc...the 9mm might make more sense.