If restricted to just one, it would probably be my McMillan stocked Brno 602 in .375 Ultra. Its reliable, accurate, versatile, and flat shooting, which pretty much defines a general purpose hunting rifle. Its not the choice for shooting beyond a half mile, and the bore diameter is a mite small to be the optimum professional hunters rifle in Africa or Asia, but in the role of a general purpose hunting rifle, there is no land animal on the planet that it would be undergunned for. Provided appropriate bullets were chosen, and shooting was kept within 500 yards, it would do its part. It jumps around a mite, but shooting it doesn't upset me; as for coyote shooting, I bet its easier on pelts than a 6mm loaded with varmint bullets.
Another option though is to have a custom switch barrel rifle built, with a bolt body for each family of cartridges, so you can pick the optimum cartridge, barrel length, and barrel contour for any type of hunting you might consider. If you want a .22 centerfire for coyotes, you're good to go, a .280 for deer moose and elk, is at your finger tips, a .338 Lapua will let you reach out and touch something, and your pals at the range who think a .300 magnum is a big rifle, will be highly impressed with the .585 Nyati. That's pretty versatile for a single rifle.