If you can get it , l would suggest a .22 Savage High Power , also known as a 5.6×52R. My very first hunting rifle was a gun in this Caliber. It was a Savage Model 99 lever action . One of those older models with the 5 round rotary mags. That is probably the most accurate lever action , l have EVER seen. The gun was a literal tack driver. I used to load the gun up with 80 grain sisk bullets and it was a beauty. Back in those days , you could legally hunt a deer with a .22 centre fire in Maine ( maybe you still can , l don't know. It's been many years ) and l took 14 deer with the rifle and it never failed me with neck shots . It also took down 200 pound Feral hogs , coyote rabbit .
The gun's previous owner ( my friend's dad ) even took a black bear with it using a lung shot ( of course l wouldn't recommend doing that with a .22 anyhow , but it's reassuring to know it can be done ) .
You can find old BRNO combination guns in this Caliber , second hand , online or you might also come across the occasional Model 99. Take it . You won't regret it.
Ammunition ( as of 2010 , as far as l know ) is still made by Sellier and Bellot .
I still regret selling mine.