I had a nice size rifle I believe would fit your described needs, the Remington Model Seven in 243. It's not a long barrel and solidly built. However, it is loud and the 243 round has some recoil. I shot groundhog and whitetail deer with her, so I'm sure it would take a coyote. However, a new one is probably over your price limit.
Here:
http://www.remington.com/product-fam...del-seven.aspx