You said "varmint rifle" without specifying varmint...
The .25/'06 and .270 WCF both date from the days when Uncle Sam left empty brass laying around (FREE) and surplus ammo could be had for $0.01 each or so... Shove the empty case into a die and go... Long, respected great reputations...
At the same time, no one calls the .25/'06 efficient. Velocities can be equalled with all but the heaviest bullets and slowest powders in smaller rounds. I think it was Mr. Titus who did the .270 Titus, essentially a .27/.250 Savage... If you are thinking coyotes, this is very bearable. If you are thinking p'dogs... barrel might last a half day, very good day...
As is often pointed out, there are tons of 6mm match bullets; not nearly as many match quality .25 bullets. More recently, but still a deficiet.
The options are endless and encapsulate the history of the 20th century. .250 Savage probably began it all with the "new" velocity of 3,000 with an 87 grain bullet. !! .257 Roberts was another (7x57 necked to .25). Post WW II the .243 Rockchucker came on strong (.257 Roberts necked to 6mm and more shoulder angle)... Then the .243 WCF hit and Remington tried to compete with the .244... (now 6mm Rem). I believe Mr. Ackley thought that the .257 Roberts Improved was about perfect for the bore with reasonable barrel life. What is your pleasure? (Read: how much do you want to spend?) .250 Savage necked to 6mm is the 6mm International, very respected match cartridge and available, last I looked, in the 40X Remington... (been a while).
benchrest.com has the top rifle smiths going. They just don't go too fast. Lots of work. Proven on match ranges. But how long you want to wait? Jarret. Holland. Many others do the work as well and quicker delivery... your bucks, your call.
I will never forget an article in Sports Illustrated. They were reviewing the long range boys in PA who shoot at 1,000 yards. And most guns were monsters, bedded into railroad tie sections (or the new thing, aluminum I beams...) . AND they said that to "play" on days when the wind wasn't bad, they would get out their .25/'06s and compete ...
I would be inclined to the 6mm International. Just neck up your .22/250 brass... but I tend to be different, too different? Or, so you could easily sell the gun when the barrel had a bit of wear, .243 WCF. Find a deer hunter and make him/her a deal. .250 Savage hangs on. .257 R's are still out there. And the .260 Remington just might become as well thought of in USA as the 6.5x55 has in Scandanavia... Your bucks... luck.