ORIGINAL: North Texan
My personal preference is the .223. If your buying factory ammo, it is relatively cheap and easy to come by. Black Hills makes some great loads. The .22-250 is also a great gun. I usually hunt areas with cover during the day, and the more open areas at night. Seldom can I see much past 100-150 yards, so most of the shots I am taking will be in that range. At 150 yards, any .22 caliber centerfire will do the job. I've used Hornady V-Max's, and hide damage has been minimal on healthy adult coyotes. I vary between 40, 50, and 55 grain bullets depending on the situation. The 40 gr. V-Max's work great, and even at fairly close ranges I haven't had them exit. The only downside is I have made some shots at longer ranges where I hit my mark, but for some reason or another, the critter didn't go down. I haven't had that problem with the 50 grains or larger. The effective rangeon both of these is usually further than the majority of shooters can take advantage of.
If you are needing something where you will frequently be taking shots past 300 yards, then you might look into something packing a little more heat, like a .243 or a .25-06. These will reach way out and touch coyotes, but they tend to damage the hides a little more (especially at close ranges), and ammo can be a little more expensive.
Just my $0.02 worth.
DITTO!