Depending on the animal, you might have a tough time with that 38 Special.

Here's one for you on 38 caliber handguns and skulls -

Everyone knows that black bear are small, light, and easy to kill, right?
Head shot should be easy, right?

38 Special should be enough, so 357 Maximum should be even better, right?

Siskiyou county, fifteen or so years ago, bear season.
Some lucky dude gets his tag, finds a top quality guide, has excellent gear, strikes it good on hunting day.
Really nice big boar, 357 Maximum loaded for bear.

Dude center punches the bear right between the eyes.
Bear goes down.
Dude jumps in circles in celebration.
Bear gets back up and clobbers dude, knocks him to the ground.
Guide saves the day and kills the bear.

Hunter ended up being ok after all.
As they examined the bear, that 38 caliber bullet was found pegged to the bear's skull, right where it hit. The bear's skull was creased and split, part of the bullet nose had extruded into the split in the skull.
Might have eventually killed the bear, but not quite right then.

I'm responding to your language of *Stopping*
I'm doing this in a kind and generous spirit in order to shine some light from the field experiences of some very determined hunters and shooters.

Telling yourself that you're doing "Stopping" business - you might end up convincing yourself of that, where maybe you shouldn't be.