What instrument or instrument-parts am I looking for?

I'm looking for something like a chromatic saxophone but without too much bells, metal, and keys. You know how you can put a saxophone's mouthpiece around a trumpet's lead pipe and change what octaves you play by covering the mouthpiece? Well it's really hard to cover it with my entire palm while holding the trumpet and pressing the keys. If no instrument exists, is there something I can buy? Something that makes it easy to keep the sax mouthpiece in place, around the trumpet, without having to use my whole hand to change octaves? Is there a name for this? By the way, the pitch changes depending on how much air from the mouthpiece went through the trumpet and how much air escaped around it's gaps. I think of it like a diaphragm of a microscope.