Which barrel do you already have?
I generally recommend the .30-06 if you're going to have ONE caliber. You'll be overgunned for coyotes, slightly undergunned for brown bears (which most of us in the lower 48 will never hunt in the first place).
And, in all sincerity, many of us end up blinded by "big bears" when we're choosing a caliber. Chances are that if you can afford a guided brown bear hunt, you can also afford a dedicated brown bear rifle. Otherwise, encounters with bears are likely going to be blacks (which will fall to a .243), or grizzlies (which are considerably smaller here in elk country than they are in coastal Alaska/Canada - and will fall to a .30-06 or .300)
If you can have TWO calibers, I recommend the .243 on the low end for everything through deer-sized game, and something in the .338 class for elk and up. You'll be a shade overgunned for coyotes.