Each critter will be different on the release technique.
I use either a choke pole or a piece of plywood with a V-notch cut on the bottom. Sometimes, if the critter is extremely feisty, I'll use both.
Canines get the catch pole.
Possums get the V-notched plywood or just a stick across the neck.
Coons get the V-notched plywood.
Cats get the V-notched plywood with a blanket or coat thrown over them.
Skunks I get the neighbor kid to release them. He has a knack for doing it. lol
I like the plywood because it keeps you and the animal separated. It can be also used to push or direct the animal away from you after you release it.
One thing I never do, even with the plywood, is release an animal by hand. I always use my feet. Just a personal preference, because I don't want to get my face or hands near their teeth. I'll let them chew on my boots all they want, but not my bare skin.