That's a tough one. I would want to use something around a 30-06 for the elk but I'm not so sure I would want to use that for a coyote. That's a bit of overkill.

.308 is big for coyotes but it is very accurate (which you need for coyotes). It is also big enough to bring down an elk from a ways out.

How about a .50 BMG...

.30-06 is the best way to go. Its certainly overkill for a coyote, but I'd rather have that than shoot something that might not humanely kill the elk.

I hope you don't want to save the hide from the yotes though.

That range of versatility is difficult to limit to one cartridge, but the 7mm-08, the .25-06, the 6.5x55mm, the .270 and .280...all good choices. I'd include the .257 Roberts if you handload, and the .308 rates highly. Fortunately, you have good cartridges from which to choose, but the difficulty lies in choosing one (1) rifle & cartridge!

The .270 would be hard to beat. A .308 would be right there too. If I were doing the smaller stuff most of the time and then occasionally hunt somthing bigger I'd lean .270. If you were going to mostly hunt bigger species go .308 IMHO.

I agree. I've seen .30-06 rounds run from a 100-grain "plinker" to a 220-grain rated for large bear. I have several types always handy, and just match my rounds for the job.

.50 bmg because you dont have to gut them all you have to do is shake out each end

Not as many people say so, but I really think the best cartridg for that range of animal is 44 Rem Mag. I use it out of a Marlin rifle on deer and it just flattens them. My Bishop at my church uses a Freedom Arms 44 mag revolver on elk and has never needed more than one shot at one. It just flattens them too. In addition to it's knock-down power, 44 mag is good for coyotes and other small game. I really like this round!

.270 is one of the most popular elk cartridges. That's about smallest I'd comfortably use on elk and about as big on coyotes with any hope of saving the hide.

Too slow? The '06 shoots 150 grainers faster and flatter than a .270. Remember that a .270 is just a .30-06 necked down.

By the way, the smallest recommended cartridge for Elk is the 25-06!

I have to say the .270

If you want to use the coyote pelts, small is the way to go, even a shotgun would be better than a single projectile, as multiple small holes are easier to conceal than one big one.
If you just have to have one rifle for everything, I vote for the 30-06. Big kill power in 180 grain for elk, and Remington's "managed recoil" is 125 grain with less powder, which emulates the 30-30, and that means limited range. I know there's factory loads smaller than that, but you're giving up bullet length or bullet diameter (sabotted), and thus stability, so it still seems to me you would do better to get a smaller caliber firearm for the coyotes.