If I was building a dedicated coyote rifle, chances are it would be a .243 or a 6mm Remington. There isn't much I could do with a .22-250 that I couldn't do as well with a .223, and while I like the .22-250, its advantage isn't as obvious as it was a generation ago. If you live where the wind blows, you might find that the sub bores are significantly more wind sensitive, then the heavier .224s and 6mms, and some might even opt for a .25-06 or even a .257 Weatherby. The bullet options that are available in .224 and 6mm makes the .17s and .204s nonstarters for me, but for some people a wide selection just confuses the issue.
A bigger issue than caliber is whether you should have a varmint barrel or a sporter weight barrel and what rate of twist it should have. If you're a hiker, a light rifle can be an advantage over the heavy barrel version. Game is not shot with groups, and if only the first shot or two hits point of aim, that's all you need in the field, but a sporter weight rifle that is properly bedded is better than that. Some argue that the best accuracy is realized with the minimum amount of twist needed to stabilize a given bullet, others argue that a custom rifle should have a twist that is chosen for a specific bullet. I disagree with both. I like fast twists, and over the years have pretty much debunked that idea that a slow twist has any significant advantage over fast twists. A 1:7 .224 will shoot 50 gr bullets just as well as a 1:14. If your rifle barrel has a fast twist, the terminal performance on yon coyote will be much more dramatic then you might expect.