Recently, I have been having almost a breakthrough with my trumpet range. For the past month or so, I have been consistently able to hit an E/F above the staff everyday. However, there is only a short time after I’ve warmed up (that i like to call my prime time) where I can play these notes strong and bold enough to be deemed actually playing them, until my playing becomes queasy. I got the job of playing lead trumpet for my high school jazz ensemble this year, and I’m afraid I won’t be able to play those high notes like a lead trumpet, consistently throughout an entire concert. I realize all I really do is play double octave scales to practice, which I guess is not really much of an endurance exercise for the upper notes, so what kinds of exercises or practices would be suggested for me?
Playing high is much more about control than strength. You're probably getting tired quickly because you're trying to rely on strength rather than air support and a controlled embouchure.
The best way to improve range is not to work on playing your highest notes, but to work on making slightly lower notes easier. Focus on trying to play high C with as little effort as possible, supporting it with a fast airstream and firm but controlled corners.
Working on pedal tones can also help. Counterintuitively, the same kind of control needed to play high is also needed to play low. I really like the Stamp exercises, or you can simply play scales that end below the standard range. Pedal notes like to wobble like crazy, so the challenge is to hold them steady using a focused airstream and holding the corners perfectly still.
Here's a different slant—try being a team member. If you know you have an exposed passage coming up, arrange with one of the other trumpet players that they'll do your line for a couple of bars (which will be in the stave) while you gather yourself before coming in above the stave. Also, farm out the high parts so everyone has a go. This is much less pressured for you, and everyone else in the section gets the opportunity to shine.
It is normal that you can reach the higher tones easier when the instrument is cold. Further your lips and muscles are getting tired - also from warming up.
I've seen a video where a trumpeter puts his tongue between the under lip and the teeth to play high notes. The tone is not so fine but at least he is able to reach the notes.
When I am not in form I try this trick.
As you know there are many different embouchures and also many bad habits, like e.g. to blow up the cheeks to get the tension, but in an exception situation everything is allowed ...