If your trumpet mute keeps falling out (like mine often has), the problem is probably that the cork is too dry. You can temporarily solve the problem by wetting the cork slightly.
One suggestion I found online was to put the entire cork end of the mute in your mouth for a few seconds before putting it in the trumpet… Hmmmm…
Someone else suggested breathing on the inside of the bell before putting the mute in – like breathing on your glasses to clean them.
Another more permanent solution is this, found at trumpetherald.com, posted by Tim Wendt in 2013:
“Get a piece of bicycle inner tube from a mountain / trail bike. 1 3/4” is the size you want to go for. A street / road bike tube will be too small in diameter (although I did use that size for my piccolo Harmon mute).
Cut a section that is about 1 1/2 inches long. Wash it out (it will probably have powder on the inside), let it dry.
Now for the hard part – spread it open & work it over the cork on your mute. This is easiest done with the mute set on a sturdy, flat, non-skid surface. You might need to work it down with your fingers. It will fit properly when the tube wraps around the edge of the cork (where the cork is at its widest).
It’s important to leave the cork on the mute. This will not only provide a surface and edge for the tube to grip onto, but it will also keep the mute a proper distance from the bell.
Trim any excess tube from the mute hole.
I don’t use any kind of adhesive. I find the natural grip from the rubber trying to get back to its original size more than enough to keep it in place. On the rare occasions where it starts to slip off, I just work it back into place. It has never “snapped” off.
The rubber provides a solid grip in the bell, and it’s also easier to remove when you’re going to “open.”
I have done this to all my Harmon & Bubble mutes, and in the last 3 years, they have fallen out maybe 2 or 3 times total for all.
I’ve also done an A/B test with a mute just cork & mute cork with inner tube. To my ears and the ears of others who were listening, there was no change in sound. We were a bit surprised, as you’d think rubber would dampen the sound. But to our ears, that proved not to be true.
If you don’t have a bike or friends who do, inner tubes are very inexpensive and will give you more than enough material to do all your personal mutes, the mutes of your section mates, the mutes of complete strangers…”