It doesn't have to be big enough to hunt them, it only needs to be sufficient for self-defense; that is, short range and big enough to do the job within it. I wouldn't go with anything smaller than .38 special but you don't need to be toting a cannon either. Just my $0.02.
We actually have several hand guns and several shotguns... including a 410 that we got specifically for snakes.
When we first moved here it was the middle of nowhere, and we always walked armed at night. Now it is pretty built up, but still with lots of woods and space. (too many houses to hunt here, but the houses have 5 to 75 acres)
The coyotes have moved in the last several years, and we figure between the late freeze this spring and the drought this summer they may be short on little critters to eat.