When I ran a trapline I carried a Marlin Model 60 .22 to pop the coyotes in the head. A .308 won't be very kind to the fur if you want to have a nice pelt. Entrance hole might not be too bad, but the exit could tear it up a lot.
In Georgia a rimfire is required when checking traps. It is one of the most fur friendly rounds, but I'm guessing he was thinking of hunting instead of trapping. A .22 can still get the job done, but you need to be a good caller and good at camoflage to get them in close enough to be real effective with a .22.
Also why I asked about distance and species. Popping beaver, otter, foxes etc. in traps is a bit different than calling in a yote or a bobcat. Two different skill sets. Getting them close enough to shoot is one thing, getting them to stem inside a 3-4 inch circle is another. Both take a lot of knowledge and prep, but the tools can be a bit different.