There seems to be an interest in coyote calling here so I'll drop my $0.02 as to what works and what don't for various outlooks.
1) if you don't care about the hides (any $$ helps offset the cost) ANY centerfire works. Use your deer rifle with your deer load, you can use the practice.
2) in tight cover with fast shooting, a 12ga auto with lead BB or steel T in a full choke is good, very good. BBB might be better (lead). 10ga is a better "stopper", but the 12ga handles better overall. IMO, the cat's meow is a Browning A-5 3" with a barrel cut to 20" and choke tubes.
3) saving fur - a CENTERfire 17, 22 Hornet, 218 Bee, etc work the best. You can fiddle with 223, 22-250's and such, but the 22 Hornet and 218 Bee are cheaper long run and the 17CF's are the ticket. The 204 Ruger looks promissing, withholding judgement until the 30gr bullets get some field time.
In summation, if you want the cheapest fur friendly setup, buy a NEF syn stocked 22 Hornet. Get 1 can each of OD and med brown krylon paint. Mount a 1-5x scope (I have a 2.5x fixed on my calling gun). tape the lens/adjustments, gaps around the hammer/trigger, then paint the sucker some cammo pattern.
But.....ain't many troubles that a man caint fix
with seven hundred dollars and his thirty ought six.
Lindy Cooper Wisdom