The .25-06 would be an ourstanding deer cartridge, and will do a great job on varmints and predators too, insomuch as it will make any varmint or predator (short of the great bears, of course) VERY dead!
The only two problems I could foresee would be that the .25-06 with varmint bullets would be pretty hard on pelts. The solution would be to use a heavier constructed bullet that wouldn't expand much or fragment in a coyote, leaving a small exit hole. Just be careful (as always) about what's behind the target as the bullet will, unlike a varmint bullet, pass through with a lot of energy left and could ricochet.
The other problem you might run into if you do a lot of high-volume varmint shooting (prairie dogging), where lots of rounds are being fired. The .25-06 burns a lot of powder and barrel heating would be an issue. But if you just used it to for the long shots and had a .223 or .22-250 for the bulk of the shooting, then the .25-06 would be great.
As for overkill, the .25-06 isn't THAT bad.

I met a guy the other day who takes three rifles with him when he goes to Dogtown, SD. One is a custom Mauser in 22-250, one is a semi-custom Remington M700 in .22-243, and the last is a 7mm Mag. He loads the 7mm with match-grade bullets and uses it for the occational 400+ yard shots! Talk about really ruining a prairie dogs day!
Mike