For putting a down but not out critter out of its misery my most carried gun over the the years has been my S&W M63 .22LR. That has been mostly carried in an El Paso Saddlery Tortilla Pancake. Rides comfortably unnoticed until needed, gun is high and doesn't hit the ground when sitting and can be under a jacket out of the weather or brush.
A larger gun for hunting or defense gets packed in a chest rig. Not a shoulder holster, a chest rig. The Guides Choice by Diamond D Custom Leather works fantastic for my G20 10mm.