Teflon tape is your friend.

Have you ever had the experience of being sharp because the warmer weather and trying to pull out your mouthpiece only to find it so loose that it wobbles and barely stays on the neck? Then you tried to use some strip of paper as a temporary fix to wrap around the neck only to find that it was difficult to put on the right amount for the sax to be in tune. I have found that using Teflon tape works great and will actually form fit the extra gap and still allow you to make micro tuning adjustments without the mouthpiece sticking to the neck and causing the cork to crack or break off. You just take a piece of Teflon tape approximately 9 inches (use more or less depending on the gap) and starting at the tip of the neck wrap a few layers slowly spiraling away from the tip. When you slip the mouthpiece over the neck the Teflon tape will compress to exactly fit the diameter of your mouthpiece. Since there is no adhesive on the tape you will have to hold the first wrap in place till the tape overlaps with itself. I have actually used Teflon tape to completely replace my cork on an older baritone sax neck. The practice of keeping a roll in my case has bailed me out on my baritone, tenor, clarinet, and bass clarinet. –Garrett