A VERY GOOD QUESTION ...
This one concerned me too, so I asked a veteran pigeon shooter who told me an Excellent QUICK solution !!
Hold the pigeons head in your hand, placing thumb on top of its head
AND PRESS DOWN HARD INTO SCULL with your thumb.
It dispatches quickly and keeps pigeon in good nick to be used in decoy pattern !!!
All pigeons, especially young ones have a soft top of the head.
I find this quick and the least distressing for the bird.
ALTERNATIVELY, get close and shoot it again, but a bit messy !!!
Dudemonkey