For longer range deer and coyotes, out to 600 yards, its very hard to beat the 25-06 w/115-120g bullets at 3100fps. I really like the 115g NBT with its .450 BC. I've dumped a few deer and coyotes between 500-600 yards. IT works very very well for this purpose. 25-06 isn't the greatest elk cartridge, but with a good bullet and shot placement it will get the job done to 400 yards or so.
I"ve never owned a 260, but if I did I would shoot a 120-130g bullet. It will knock any deer flat on its butt to 500 yards or so. If you wanna go after elk, you could load a good 140g bullet and be perfectly suitable for elk to 300-400 yards as well.
I owned a 270 for a short time, I loved it. It was a remington 700 BDL and shot lights out with 150g sierra gamekings with Rl-22 or H4831 at 2950fps. It will take deer to 800 yards and elk to 600 w/out hesitation. I wished I never sold the gun, like I said it was shooter with those 150's. It hits a lot harder then the other 2 past 600 yards with the 150g bullet. More suitable to elk then the other 2, great long range deer rifle.
Its a toss up between them all, but if I wanted a dual purpose coyote/deer getter, IMO I dont think theres much better then the 25-06. For coyotes you can use the 85g NBT at 3600 and watch fur fly, or go to 100g NBT at 3350fps for coyotes/deer as well. If you want the longer range capability, the 110-115g AB/NBT at 3100fps will do anything you want to 600 yards. The 25 doesn't kick hard at all, much like a 243 IMO, its very acurate cartrdige, fired my smallest 3 shot group in my life at 100 yards with my Winchester classic in 25-06, it measured .045". I've killed everything from squirrels to black bear and my sister wacked a cow elk at 300-350 yards last year.
I pretty much wanted a medium sized caliber that was fast, flat, and accurate with some punch for deer size game. I didn't want the 270 because you couldn't get anythign lighter then a 130g NBT, there wasn't much to choose from in the .264" department, I dont really care for 6mm much for LR deer unless its set up accordingly, so that pretty much left me with the .257" caliber and the 25-06. That was about 6 years ago. I've already shot 1 barrel out, and I"m on another. I"ll always have a 25-06 around, its a pretty versatile cartridge. If I didnt like the 25 so much, I'd still take a 270 anyday. I've never been a real fan of shortactions, so that kinda leaves the 260 out for me in the end...