Greetings from the Great State of Georgia,
I am a knowledeable predator hunter so I feel that I may have some advice from a different perspective. If pelts are to be saved a traditional small caliber is a good choice. A Handi in 22 Hornet is quite effective out to about 200 yards. A soft nosed bullet will do the job without destroying the pelt. The ever reliable .223 will do anything that needs to be done with a FMJ or "regular" bullet but a Ballistic Tip will sometimes put a large ugly hole on the backside. The .243 will add some range as well as a bit of stability an longer shots where wind is a factor. Again if the pelt is to be saved care must be given when choosing the bullet. A .243 will definately dispatch a small predator but will also blow a H U G E hole in the pelt with the wrong choice of projectile. I think the largest varmint bullet would be in the .257 range which is available in the Handi as a 25/06. This is a good choice for long range shots using lighter bullets while being a good choice on medium game using heavier bullets.
Now for a completely different angle. If pelts are not to be saved, any gun that is accurate at the intended range is a good choice. I can think of no better choice for nailing a coyote than a full max load of 400 gr. 45/70. There will not be much tracking involved !!! Other good choices are, .270, .280, 30/30, .308, 30/06, .44 mag., .500, or maybe even a .50 cal smokepole. My all time favorite varmint gun is my .308 with FMJ simply because it is extremely accurate out to 300 yards and beyond. When hunting at night and counting on up close and personal action a good shotgun is an excellent weapon. I have literally cut coyotes a backflip at 40 yards with my old H&R 10 GA. shooting 3 1/2 inch #4 shot. Nowdays, the 12 GA 3 1/2 is almost as potent and much more versitile.
Hunting predators is a great wat to stay in shape both hunting and shooting. I guess the most important factor in choosing what weapon to use is to decide exactly what you want to do and go from there. I would definately suggest a centerfire over any rimfire. While a .22 Mag. or .17 HMR will dispatch a small predator, there performance downrange is easily surpassed by even the dimunitive .22 Hornet. Good Luck and keep us posted on your choice of weapon as well as any success using it. God Bless !!
Dave