I have always recommended "ear hole to eye hole" method. Having guys imagine their bullet passing from the close side ear, to the opposite side eye.
A .22 will (and does) work with this method.
The problem(s) is that in the heat of the moment most guys forget and start blasting away.
I later learn that:
They forgot the advice and try to shoot through the skull, which often gets deflected.
They were trying for "heart" shots.
They are decent pistol shots on the range, but have a harder time in the field.
The animals, while injured are still semi mobile, and not presenting a stationary target. That happens in a lot of cases. They get one broken leg, and their survival instinct kicks in, so they won't let anyone close. They will stay at least 50 feet away from people as they consider them a threat.
If you have a rifle with appropriate ammo that you believe will penetrate into vital organs, then this would be a good time to use it. If not try to position yourself to set up for a clean head shot with a solid backstop.
I stay away from neck shots.
The other thing I have noted is that people can get worked up and start to shoot repeatedly. This is not about shooting a lot. It is about shoot well.
Just once, and turn the lights out. It is far more humane, and safer for the community.
Here is a pic from last year where a 230 grain was used to put down a cow elk that the vehicle in front of me hit, when I was on my way to a shoot with friends.
Ear hole to eye hole.
A single shot and her suffering stopped. You can see in the pic where she was struck by the truck.