One of the toughest places on a saxophone to keep silent while still feeling smooth and tight is the link from the side key lever to the pad cup on most modern horns. Here is a quick technique we use in our shop:
Remove the old material from the shaft on the end of the pad cup arm. Cut off a piece of Hot Glue Filled Heat Shrink Tube that is the smallest diameter possible.
Shrink it over the shaft. This heat shrink tube will remain more resilient if it is not overheated, so hot air may be a better choice than a flame.
Now, shrink a piece of Heat Shrink Teflon® Tube over the Hot Glue Filled Heat Shrink Tube. Assemble the mechanism. It should be too tight at this point.
With the pad cup and lever both in place on the saxophone, heat up the joint to soften the shrink tube, while at the same time moving the lever through its normal range of motion.
The hot glue inside the shrink tube will soften and conform to the forked arm on the lever. Remove the heat as soon as the key moves freely. Continue moving the key while the shrink tube cools so the spring tension doesn't force all of the hot glue to one side of the joint. After it's cool, test that the key moves freely and silently. If it still feels too tight, re-heat again while moving the key. If it is not silent, add a little heat without moving the key.
Once you have this working properly trim off any excess tubing.
Teflon® shrink tube has a tendency to slide off over time if not glued in place. Most often the nature of this joint allows the forked arm on the lever to hold the Teflon® in place, but if it doesn't just slip the Teflon® tube off halfway, put a drop of super glue under it and slip it back in place.
If you have any questions about this technique feel free to email me at: Questions@MusicMedic.com