Getting off the adhesive strip

I wanted to replace the adhesive strip because it no longer stuck so well.
It came off well on the pad itself, but not on the cylinder (see picture).
two questions:
Before you put the new adhesive strip on, you should probably remove the old one completely first so that it lies flat, right?
This then leads to the question of whether anyone has a tip on how to get the adhesive off well (I’ve already tried blow-drying it).

Rubbing it off with your thumb usually works best.

not in this case, but soap/dishwashing stuff helped