When I started hunting, my first centerfire rifle was a .30-06. Back then I was living in NW Colorado, and I used my .30-06 for deer, elk, antelope and prairie dogs.
Shortly after I moved to Montana, my new hunting partner introduced me to the .22-250, and I got one. I now have several friends that hunt almost exclusively with their .22-250s up to and including elk and buffalo, but they are exceptional shots and hunters.
To bridge the gap between my .22-250 and my .30-06, I chose a .257 Ackley Improved. It is ballisticly just behind the .25-06, and to me it is a perfect "all around" varmint to medium size game round, and with proper bullets and shot placement its not bad for elk.
Since I got my .257AI thirty some years ago, I've shot prairie dogs, coyotes, almost all of my deer, antelope, and mountain sheep with it, and made one shot kills on elk and caribou with it. It is one of my favorite rifles.