Here in CO, your "coup de grace" weapon MUST still be a legal hunting weapon. For a pistol, that would be 550 ft/lb at 50 yards with an expanding bullet and 4" barrel, not the .38 snub you stuck in your coat pocket.
When my Grandfather first went mule deer hunting here in the 50's, he shot a big buck with his .270 Model 70, and approached the downed animal to cut its throat. As he straddled the animal, the buck woke-up and give my Grandfather quite the ride, with him slashing at its neck. When the deer was finally dead, they found out he hit an antler, temporarily knocking out the buck from the skull impact.
The above incident is reinforced every hunting season in my family about how to approach a downed animal.