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If you're shooting fur for skinning, small caliber and slow velocity are what you seek. Second choice is really really small caliber and stupid fast.
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I shot a coyote that had a really severe mange a few years ago.
Used a 220 Swift pushing a 50gr V-max around 4030 fps. Bang, flop.
Walked over and there was just a few tiny dots of blood on the snow. Bullet went in through the front shoulder. Combination of speed and a v-max hitting bone and I guess that bullet just exploded. Tiniest sliver of an exit wound. Part of the jacket or the core, most of the bullet remained inside.
Shame it didn't have good fur it would have been a perfect candidate for skinning. As it was I only got a shot at him because the mange was so bad it was trying to get near a barn, I assume to try to keep from freezing. It was really cold that weekend. Probably only had 30% of it's fur left. I'm sure it was incredibly miserable.
I can't imagine a 30 cal reliably giving good pelts unless you are using 300blk with subs.
I suspect a 220 Swift with 40gr vmax or 35gr berger going as fast as possible would be a reliable fur round. I never went with bullets lighter than 50s because they shed velocity too fast, and are usually moving slower once you get out to 200 yards.
ETA:
Found a picture of it!