I utilize dewey rods and they recommend 20 cal rods for 22lr. They have a 22 cal rod labeled centerfire only. Good luck.
I wanna say the Dewey rod will still fit a .22lr, it's what I use but there are different jags I think. I have the .22 short rod and it works fine too, identical to the long one, only difference is with pistols and revolvers it's ideal to use the pistol caliber brush --it's shorter and maybe smaller and goes all the way through the barrel and frame in a J frame . Wanna say the patch loop is what you wanna use, may have to trim the patch but the jag may work, been a while. It's been awhile since I cleaned a full rifle barrel (one 10/22 is a 4" SBR now and the other is an integral w/4"bbl.). Only cleaned the integral one time, it does require the long rod to pass through the 12" suppressor tube, and pretty sure I just used the pistol brush and brass .22 loop.
They may have a dedicated one for each, wouldn't surprise me, but if the rod fits easy and the attachments as well, it's good to go. I didn't know about separate rods until just now. I do have a plethora of Dewey attachments though, it's about all I use. Though this Tipton or Hoppes carbon fiber rod that came with a rifle I got is easily as good as a Dewey and way lighter too.
So yeah, the rod fits but some attachments work better than others, just depends.
A .308 rod is usually what, .27? So yeah, a bit too big. I use a .22 rod for 6.5. A .224 rod is gonna be a bit under that so it doesn't fit tight, so that's why I say check it and see. Put a brass loop and adapter on it and try it out. Just don't force it, should just drop right in easy with zero resistance.
Another tool I use and make is to JB Weld a Dewey .22 threaded adapter fitting onto a piece of nylon coated cable I get at Ace. Get the (1/8"?) --the wire OD will be smaller than the overall OD with the nylon coating, which is just perfect for a .22. Strip back a 1/4" nylon and JB Weld. Now you have a pull through cable, just cut to length. Won't damage the barrel and it's just under the diameter of the Dewey fitting. I use 'em in field kits instead of GI rods. I make up Dewey field kits that fit in a small pouch, kinda like Otis kits. There's way's to make a T handle using this cable too. The .22 size works for all calibers up to .30.