The Savage MkII .22's are good, but have one weakness, the sear.
It's a MIM part and if it breaks (It's also used as the bolt stop so it can and does happen) it's can be a pain to remove.
There is a good replacement part, available from Brownells but costs @$90.
The older Remington 5XX series, the Marlins and older Savages can be had and as long as the barrels aren't toasted by improper cleaning should be good to go.
My 511 is extremely accurate and didn't break the bank, they are available for varied prices depending on condition. Same goes for the Marlin bolt guns from years past.
Many don't think the .22 is good at longer ranges but the .22 used to be shot at 200 and 300 yards regularly in competition. It's much more dependent on good ammo and a good shooter at that point. My Kimber's and Volquartsen 10/22 are reliable shooters at long range, the Volquartsen is good for under 4" at 100Y from a rest and with several types of ammo.
The Kimber is good for half that easily, but the ammo is the biggest telling point here.
Shoot garbage and you're gonna get garbage results.
Think CCI Green Tag, Eley Tenex or some other target (subsonic) velocity ammo that's not bulk packed.
I've used the lowly .22 on prairie dogs out to 200Y, with great effect. It takes a bit of practice to learn to dope the wind and estimate the ranges correctly to compensate for bullet drop at longer ranges but once you learn, you will find it a really fun way to spend an afternoon (day).
Sam Fadala's
"Book of the .22" is a great read and well worth searching out. It has a chapter on long range .22 shooting that's well worth a read.