Nice war starting up!!!!
Keeping the original question in focus....I have had or have an Anschutz target rifle (single shot), 10/22, Savage Mk II BV, Remington 597, Savage 93GL (22 Win Mag), Savage 93R17 BVSS.
The last two are magnum rimfire guns - both priced reasonably low and shoot great! Either could be a good option if you want a hunting rifle with more punch than a 22lr - and - also good on paper. They're all ammo specific - every rimfire has it's preference. You have to experiment to find the ammo yours likes to eat.
I'll give my rank for the listed 22 LR's for accuracy with comments:
#1 Anschutz - hands down THE most accurate 22 I have ever owned - one small ragged hole at 50yds. It's a single shot and not a hunting gun at all. The trigger is AWESOME at 10oz. You get what you pay for and these are in the $1000+++ range (rifle only - no sights).
#2 a tie between Savage MkII BV and Rem 597 in accuracy. I'd pick the 597 out of these due to lower price and as a hunting gun, a semi 22 is way more fun. I have to say I did put a VQ hammer and extractor in the 597 to improve the performance - without the upgrades Savage gets #2 spot - otherwise, it's a tie...Still, if cost is a factor, the 597 WITH the upgrades is better value than the Savage.
#3 10/22 - a distant 3rd place. Mine was a POS for accuracy. Fun to shoot - lots of mods available if that's your thing. Some guys swear by them - not me.
Good luck ! SD