Too many variables possible for me to pick just one rimfire maybe. While I have a specific order of priorities when carrying a rimfire handgun, it still depends on the parameters of the hike. Short or day hike, a weekend or if for a week or two into the back country. And terrain. Now days lugging a big, heavy handgun over tall mountains or across hot deserts for days, weeks would not be a first choice if I had something I can just stuff in a back pocket that will get it done just as well.
In most cases when I carry a rimfire handgun hiking it's primary duty is for cook pot fodder, but also just for the fun of shooting it or for better than having nothing protection. Which RF handguns were carried have been quite varied during my life with lots of them, as well as including revolver, auto or singleshot. Those that were favorites were carried the most though - easy carry, reliability and just plain how well I could shoot them myself prevailed when I got older.
Some favorites I have carried....
Ruger Bearcat, Super Bearcat or Single Six old models.
Colt New Frontier Buntline convertable
S&W 22/32/34/63 Kit Guns
High Standard HD Military
Colt Woodsman/Huntsman
Ruger MK I, II
Browning M41
Colt Ace
Beretta M21 Bobcat
Astra Cub 22Short
Stevens 35/43 Tip Up
Stoeger Tip Up
H&R Handygun
TC Contender
I have scaled way, way down on all my firearms the last few years, so now my rimfire handgun choices are limited to 4... yet all of them are very capable for all of my uses. In order of what would be my choice to carry now days...
Beretta Bobcat 22LR - most of the time.
Ruger Super Bearcat 22LR old model - the rest of the time.
High Standard HD Military 22LR - seldom - too big/heavy.
Ruger Single Six 22MAG old model - seldom - don't need Mag.
If I still had a S&W Kit Gun it would be #3 - I can shoot the Bearcats far better than them even without sights. But they are an easy carry back pocket, and very capable.