The .243 was designed for the exact situation you are describing. It is a "Varmint/deer size game" cartridge. A lot of hunters would prefer a .223 or 22/250 for coyotes and a lot of hunters would prefer a .270 or something for deer sized game.

A 257 roberts or a 250/300 Savage would also fill the bill for you.

Robin down under ; where they build a .25/303 as a popular dingo, kangaroo cartridge.