Just about any "deer round" will work on varmints. That has a couple of advantages...you don't have to drop the dollars on a special varmnit gun (but what's so bad about that??) and you can keep shooting your deer rifle in the off-season to stay sharp. However, that is not the point of the question: I have three cartridges I am partial to for coyote hunting...the .25-06 with a 117 grain hollowpoint bullet...it's hard on the hide, but harder on the coyote.
Next would be the .243 Winchester...a great all around cartridge and deadly on coyotes out past 400 yards with a good load. Again, hard on the hides if you skin the songdog and sell the fur. Probably the most accurate round that I use, at least in my guns. Last is the .223 Remington in a single shot NEF Handi Rifle. The rifle is about as "cheap" as you can get for a single shot, but performs like a high dollar bolt gun. If you reload, and use military cases, it doesn't cost as much as .22 mag ammo and is much more versatile and powerful. Even if you don't reload, there is a lot of .223 ammo (mostly out of Russia) that is cheap and accurate. If you just want to kill coyotes, the .243 would be the round of choice for me. If you want the challenge of the long range shot across a 1/4 mile pasture fenceline, then consider the bigger .25-06. My personal range on the .223 is between 150 and 200 yards. For a real challenge, try calling the coyotes in close enough to use a handgun....at 25 yards just about any of them will take down a coyote. Hope that give you some ideas to think about.
If you are looking for an excuse to put off buying new shoes for the baby and get another rifle instead, you're on your own!

We're on a mission from God.
Jake & Elwood Blue