For 12 gauge guns, I use an aluminum rod and an old 20 gauge bore brush. The brush is "rolled" in 0000 steel wool, the whole affair dipped in a bit of Hoppe's #9 and then the tube is scoured out. Makes for a mirror shine lickity split.

Then, Naki, I actually use a very similar method to you!

I take the brush off the rod and then take a 1/4 sheet of paper towel and roll it into a ball and shove that thing thru the bore. It absolutely sweeps the bore clean.

Then I oil with a patch or "mop" on the jag.