The .38 Super shares the same drawbacks for hunting that most service class semi auto rounds do. The power is marginal for big game (not enough in many states) and the common bullets aren't well suited for game, large or small.
Absolutely can be used where permitted, and will be effective if the shooter is, but not as efficient as other things.
Frequently service class guns lack the fine accuracy to make good small game guns, and bullets meant to expand in people rarely do in small game, and the bullets shaped for feeding in a semi auto lack the SWC shoulder (and often the flat nose) common to revolver rounds.
also, particular to the .38 Super is that older guns were chambered to headspace on the case rim, and are often not as accurate as newer guns chambered to headspace on the case mouth.
In the early 70s, when I was learning pistols, Dad had a Colt Govt model in .38 Super that I often got to use, provided I used my own ammo. Guess who learned to load the Super?
I was loading Hornady 115gr JHP at a time when the only factory load was the 130gr FMJ. Potted the odd squirrel or two, and a coon and a skunk, if I remember right, can't say how the bullets behaved, never recovered any, but they did work. Never considered it for deer, though it was legal where I lived. There just wasn't a good "deer bullet" to run in the Super (or the 9mm Luger).
Probably something decent or maybe even good as a deer bullet today, but I haven't looked, I have better guns for that, or for small game these days.