This happened to one of my older altos. Rather than trying to save the cork, I carefully removed all of it and then wrapped the tenon with waxed dental floss–taking care to layer the floss evenly (just like sewing thread is wrapped on a spool) and not get it too thick. Once I got the proper fit, I just clipped the end of the floss and laid it in place. The waxed floss has stayed put, has provided as good a seal as cork, and has never needed to be greased.
