I haven't used patches for better than 25 years now for rimfire or centerfire, and only cut patches wrapped around a wool-mop for shotguns. Instead, I prefer "dedicated" caliber size felt cylinders that are pulled through a bore using weed-whacker line.
Most all of the good synthetic solvents these days don't require scrubbing of a .22 rimfire bore any longer, just need a soaked felt plug, pulled through, let it sit for 15 minutes, or so, and then a couple of dry plugs to pull the gunk out the muzzle end. They do also make a felt plug with a bit more of an aggressive attitude that has tiny brass ribbons imbedded that will help with stubborn lead removing:
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