Not to be rhetorical, but any thing new or used for that price will be great, except Ruger, avoid ruger. Marlin is generally considered the cheapest and most accurate, out of the box, whether it is semi auto, or bolt. If you wanna lifetime, fine bolt action that has rediculous accuracy and a single set trigger, get a CZ.
anything savage, or remington or winchester will be fine as well. Again, new or used, you cannot go wrong with any of the listed above, or any other brand as well; I would not hesitate to pick up an old mossy or H%R either.
22 mag is out of the question, too expensive, with questionable accuracy mostly, unless you can find the mag in an old BRNO, mossberg chuckster, kodiak , or H%R, next would be an old marlin, 7 or 8 series, preferably a 7.
you call or wait on a coyote to come within 75 yds, and the 22 pill will have about 90 lbs of energy; if you hit it in the head, neck, or top chest, it will drop like a bad habit.
and yes a 17 hmr is even more flat, and has about 150 lbs of energy at 100 yds, and is going as fast at 100 yds, as a 22 mag is at the muzzle, but I hate the thought of 13 dollars plus tax, the cost of rimfire ammo.