I use both your cartridge choices on my buddy's trapline and for some coyote calling. I use FMJ bullets in the .22 magnum for headshots on small furbearers like muskrat, but use standard 40 gr. hollow points for beaver. The hollow points don't exit on beaver shot in the body and they don't mess up heads too much. They kill beaver much more consistently and cleanly than the FMJ bullets. I don't use the lighter hollow points that are so fashionable in the .22 magnum because they might not penetrate enough on a beaver body shot, and do too much explosive damage on a beaver skull. I have not used my .22 magnum for coyote, but my buddy says that a coyote requires the 40 gr hollow point bullet for consistent clean kills. They will leave a small exit hole in the fur that is easily sewn up. Use the brand that your rifle shoots the most accurately, some .22 magnum rifles are very fussy about what ammo they are fed. Mine likes CCI and Winchester, Federal and Remington not so much.
For the .222, I used to use standard 50 gr. varmint bullets with OK results but now follow the advice of a good friend who is sort or a coyote calling guru around here. He uses the Hornady or more often the Nosler 40 gr. plastic tip bullets almost exclusively for coyote. He has been doing this for 30 years, and takes between 30 and 60 coyotes per year. He also uses a 17 Rem, 22-250, and some other cartridges, but the combo of .222 and 40 gr. ballistic tip is the most effective, most fur friendly combination that he has tried. He says that only about 10% or so of the properly chest shot coyotes are anything but dead on the spot with no exit wound. The exceptions require needle and thread or some tracking. Both Hornady and Nosler sell loaded ammo with the 40 gr. plastic point bullet, or you can load your own.
