.25-06 sounds like a good choice.

no question in my mind... .270 Winchester

i would pick the 22-250 its a great rifle you can shoot all the way down to praire dogs and all the way up to mule deer  and drop them on the spot if your a good shot

My vote would be for the .243  It is a great compromise caliber for the game you listed.

Although not one of your choices I agree with the others that voted for the .243. My kids have used one for years on deer and antelope with great results and there are many coyote friendly loads.

I love the .220 Swift but even if it was leagal I personally would not carry a .22 for deer sized game. I know guys do and I have no problem with that but I just prefer not  to myself.

The 25-06 is a great cartridge also I'm just not sure about the damage to a coyote as I have never used one.

I love my Ruger M77 25-06 and it's a great caliber with a little more punch than the .243 for deer.  I love that little .243 for varmints, it's great for antelope, and will do the job on deer too with the proper bullet and choosing your correct shot angles.  Both calibers are normally very accurate and that Ruger and the .243 Sako I have are both tack drivers.  A lot of people use the 22 caliber centerfires where they are legal for deer, but I don't feel they are enough gun for any but the most experienced hunters who know where and when to squeeze or not squeeze the trigger on bigger animals.

I have  both a .243 and 25-06.  I love 'em both, and use 'em both for Coyotes, Antelope, and Mule Deer.  I use my 25-06 for Elk also.  For your situation, since you already have a 7mm, I think you should go with the .243.  A friend of mine had a 6mm Ackley Imp. he used for hunting Mule Deer down, that thing was a tack driver and VERY deadly!!  I'd have to believe that either of the 6mm's would not do great damage to the pelt of a coyote.  Look at Sierra bullets, FMJ's, for Coyote's.

For the three game animals you listed, deer, antelope, and coyotes I would opt for the .308 Winchester.

It may be a little more than needed, but I believe in having more gun than one needs if possible. Also, I like that ammo is easy to find, and the round has decent long range capabilitiy. I am a two caliber hunter right now, looking to add a third caliber in the near future. Right now I'm using the .223 and the .308. When hunting I use the .308 almost exclusively, and it has the ability to take larger game than the .223 does. While not optimal, it will work on moose, black bear, bison, and caribou.

I find it better to have more gun than you need rather than being undergunned.

Biker