If this is the tool you're talking about, the tampons/felts are for lug recesses, an yes they completely suck.
Sinclair Action Cleaning Tool Kit | Sinclair Intl
I follow Geb's method with simple patches -for the chamber body area, and use a bronze brush for the neck/chamber-end carbon buildup. Then I spray down the chamber with a canned solvent/parts cleaner. Later I'll wash the the whole barrel from breech to muzzle with alcohol anyway, as I store my bores dry-prefouled.
To aid in this I use a short length of gas line to guide patch/rod into & out of chamber without munging things up with patch-action contact. Basically a larger diameter rod guide.
I know of no good way to clean between the breech and lugs. All I know is the tampon system didn't..