As a repair tech, I use #320 grit emery cloth in 1/2" wide strips. After finding a way to secure the neck (I use a bench peg and my stomach) I "rag" the cork with the emery cloth. I like to do a few strokes with N, S, E, W on top and then check the fit.
I like to leave the cork snug and then melt paraffin wax onto the cork. I then give it a generous coat of cork grease and install the mouthpiece. The first time it goes on, it removes the excess wax. After that very little cork grease if any is needed. It is important to remember to keep the cork a cylindrical shape to match the shank of the mouthpiece.