brush is loose in flue?

Hey all. This temperature change has sparked me to clean my flue and get ready for winter. I ran into a little problem. I have 5' of 6" single wall black pipe from the stove into the support box in the ceiling. I had a new 6" double wall SS chimney installed from the roof out prior to last heating season. This is my first year sweeping the new SS chimney. I have a 6" nylon brush that fits nice and snug in my interior black pipe, but it's a bit sloppy in the SS exterior pipe. I really have to wrestle with the brush to get a good cleaning. Any ideas?

Maybe spin it "chucked-up" in a drill motor...but not too fast.

bigger brush?