That sounds like a good approach, before moving to more aggressive measures. Whether patch or brush, make sure there is a tight fit, but not so tight as to risk sticking in the barrel.
For some reason, manufacturers make .22 jags and brushes way too tight. I stick to .22 pistol brushes, even for my rifles. Even some brushes labeled for .22 rimfire are too tight for comfort. A stuck brush or jag is bad news.
