The .270win or .270WSM will suit you very well for your needs. Yes, it's going to be hard on coyotes, and it won't be legal in some states for big game (meaning larger than deer) where a 30cal or larger is required, but in general, the .270win or WSM are fantastic cartridges.

Personally, for someone living in the midwest, a .243win is probably the ideal "do-all" cartridge. We're really talking about game between prairie dogs up to lighter big game like whitetail and mule deer. For a coyote/deer rifle, there are very few choices that will better suit these tasks than the .243win (this coming from someone who definitely doesn't call themself a fan of the .243).

Add in elk country, then the .270win is probably in the running for ideal. Not too over powered for coyotes, but just enough stuff for elk, and a great deer round. No, it's not necessarily ideal across the board, but compared to say the standard answer of the .30-06, you're likely to be hunting coyotes more than elk, so the lighter bullets will be nicer on hides than the '06. Yes, the '06 would have the advantage for bullet weight and power on elk, but the pelt damage on coyotes will be a more common concern than stopping power on an elk.