Hello! Not knowing where you are at with your playing, whether you're beginner-intermediate-advanced-pro, please forgive me if I over-explain something you already know about.
There are a couple things that you may want to try in order to "tune up" the high range. One would be what I call the "On-off trick."
Start by playing a C in the staff on your mouthpiece, and while buzzing the note slide your horn (gently) onto the mouthpiece and hold the note a bit. There are a couple things you'll want to notice with this: One is that the horn may seem to want to go to a flatter (typically) or sharper pitch than you're used to. Allow this to happen--I call it "surrendering" to wherever the horn wants to place the pitch. Actually, the horn is telling you where the middle of that slot is, where you're blowing straight down the note. A result of this will be, normally, a noticeable change for the better in overall tone quality. This is the sound you want to develop or fix/enhance range with. So: buzz note of choice--slide horn on buzzing mouthpiece--hold note on horn--listen to tone--repeat.
One variation is doing a number of on-off manouvers in a row. The idea here is to match the pitch between the mouthpiece and the horn. Very often, as the horn comes off, the mouthpiece pitch will jump up. That shows a tendency to play sharp on the pitch.
Ultimately, what you're doing is learning how to play down the center of the note, where the horn is most resonant on that pitch. You're also getting a sense of really "blowing the horn" on these notes. Take it up as high as comfortable, never allowing a squeeze or strain to invade your work. Oh yeah--posture! Sit up, shoulders down, breathe into stomach, play as relaxed as possible. You know the drill.
Another thing to do would be to order Keith Johnson's "Practical Studies for the High Register" published by Harold Gore in Denton, TX. Take those studies and play them 3 times through each key--horn, mouthpiece, horn. Play your mouthpiece with finger tips -thumb and forefinger- at the end of the shank, like the picture in James Stamp's "Warm Ups and Studies." (If you start sliding up, you're pressing too hard.) That last time through the horn should be blown like your mouthpiece, generating a beautiful tone on those studies. You want to listen to yourself playing a gorgeous sound up there. This process sets you up for that success.
The common theme here is tone quality, like you mentioned in the original post. Develop range on your classical setup with your classical mouthpiece, playing a classical tone. Let your sound be beautiful and singing. Then, when you switch to your lead mouthpiece play the
same way--let the
gear brighten your sound so you cut through the mush. You'll retain much of the cool, meaty core tone that you play on your classical setup. In fact, I make it a point to physically approach lead playing much more "classically," just because it's a way to take one approach to range that can cover all styles. I'm a big believer in keeping trumpet uncluttered, sort of a lowest-common-denominator idea. Your simplest playing will be your best playing.
In saying all this, I hope I haven't stepped on any toes, that's not my intention. Just hoping there's something here that's useful to you!
Cheers!
Shermock