For many years I carried a state issued .357mag. I typically carried a speed strip with some .38spl +p SWC-HP's (reloads.... too cheap and poor to buy "factory"), for dispatching road-hit deer.
Due to lack of penetration on all the head-shot deer I dispatched, "hunting" with the .38spl ain't a great idea; even if it's legal. It'll do in a "pinch" or "subsistance" situation where it's all you've got. But they're quieter and less recoil than a full-loaded magnum, so made shooting an already incapacitated deer more practical where an over-penetrating .357 could cause a richochet hazard on pavement/asphalt. I never had a .38spl exit, regardless of shot placement though I ususally targeted heads.
I did arrest a subject one day that had taken a doe out of season, on a closed management area. (.38spl was also a specifically prohibited cartridge). I was checking fishing licenses in the area and heard the first shot. I went looking and heard the second shot 1-2min later. When I found the subject, he was dragging the doe under the fence/firebreak at the property line. He'd shot the deer with an S&W Mod14 6"bbl with Winchester 125gr +P JHP's. First shot broke deer's neck (approx 25yd shot from a portable ladder/deer stand on firebreak). Second shot was to dispatch the still writhing deer. That was a very expensive deer. Inmates got to eat it, too...
Funniest epsiode of .38spl shot deer was by my captain. He drove up one night on a car wreck/deer-scene. The Sheriffs deputy on the scene ascertained if it was "ok" to dispatch the deer that was in median of divided 4-lane that was "thashing and bleating" to the dismay of bystander/passerby's. 'Sup' said "go ahead". The deputy shot the deer in the neck with a 9mm S&W Mod 659 w/115gr JHP. Deer got up and ran into opposing lane of traffic and started kicking and bleating louder..... My 'sup' walked over to the deer and in disgust aimed his 2" S&W mod 60 w/158gr +P SWC and shot the deer between the eyes whereupon it layed down, let out a final gasp and gave up the ghost....
He was something of a celebrety for a while around the coffee pots. (it even made local news paper as a reporter was at scene). About 6mos later, the S.O. traded in their 9's and went to .40's..... also a dept. policy of NOT shooting injured wildlife.....
But in all honesty, it wasn't the 9mm's fault. It was poor shot placement... The 9's no better or worse than the .38spl....