I first mentioned this in the buckshot discussion but I think it deserves its own thread.
There would still be more that could be done to improve shot and expansive ammo, but I think this solution would go a long way toward making these ammo types feel viable in CotW.
The image below is a roebuck I shot twice with 16GA buckshot at 25m. The species is dead center in the ammo's class range, and the distance is exactly at its lowest zero, but it wasn't enough to hit vitals (the second shot didn't hit any organs at all). However, enough damage was done that it only took a couple strides before falling dead, resulting in 100% quick kill. It seems pretty silly that such a kill doesn't count as ethical in the game (particularly when you consider that a shot resulting in the animal bleeding out over hundreds of meters can pass as long as it nicked a vital). And the same applies to a shot with expansive ammo that gets a quick kill without hitting a vital organ.
The specific threshold for quick kill I'll leave to you (80%? 90%?) but I think having two ways to pass the fourth check would be consistent with the ethos of the game while also improving the experience with half the ammunition in the game.