more carefully aimed shot to the head aiming between the top of the top lip and the bottom of the nose- which would take out the brain stem.
Pretty tough with a handgun.
At Gunsite they taught to aim for the triangle formed by the two eyes and the tip of the nose.
This is a relatively open area in the skull, and penetration is liable to be better.
You do not need to hit the brain stem as long as you can get a bullet into the brain and produce some trauma, and driving a typical handgun caliber that precisely is not as high a probability event as a rifle round.
The CNS does not react well to trauma.
It took the army shooting cats in the head with a BB gun years ago to determine that respiration often stops following brain trauma, but that if artificial respiration is used to support the victim the brain could recover and breathing resume.
Before that research victims who stopped breathing after head injury were often considered dead on the spot.