Consider the .243Winchester. I'm not saying it's best for your situation, you'll have to figure that out for yourself as you are the only one who will have all the details. Just don't overlook it. Actually, consider every calibre anyone suggests, the process of thinking through what would be good about it and what would be bad about each will lead to realising which one has the least bad and the most good. There isn't "perfect" but something might be pretty close.
e.g. .243Win:
Pro: versatile - you can get factory ammo with varmint bullets or reload you own, so if you use it for coyotes you have the choice of not ruining the hides. Larger bullets are deadly enough for deer and black bear. Recoil isn't bad so it's easier to shoot accurately, which helps ensure the smaller bullets compared to .30 calibre cartridges are big enough. Availability - it's not an uncommon cartridge, you won't have trouble buying ammunition or the components to reload.
Con: Although with good bullet design and correct shot placement it can certainly kill bigger game, if you are going to eventually go for moose it would be better to have the margin of a bigger calibre. (Mind you, that's an excuse to buy another gun, so maybe it's another advantage of the .243?)