air.

Hi,
as a trumpet player with over 40 years experience I find that the best way to improve the upper register is to start practicing long notes first in the lower register.
In Cat Anderson's trumpet method (Duke Ellington big band, listen to El Gato, Jam With Sam live!) he starts with sustained tied semibreves on G (2nd line of treble stave) at a piano dynamic and then for twenty minutes (breathing where necessary) crescendos and diminuendos! This helps to build up the embouchure muscles.
When I warm up I tend to stay below a C (4th space on treble stave) and play long notes (minims/semibreves etc, sorry I am English and don't use whole/8th/16th notes etc) from C (1 ledger line below stave) to G (2nd line of treble stave) for at least 10 minutes whenever possible.
This is like an athlete warming up, the muscles must be given time to adjust/get the blood flowing or you will find fatigue setting in very quickly. If an athlete does not warm up properly torn muscles/strain occurs!
Playing in the extreme upper register will not happen overnight, if at all!
Everyone has a different set of circumstances to contend with; teeth, lips, muscles etc.
Playing very high is something most trumpet players strive to do, not all players can be like Maynard Ferguson or Maurice Andre. If you can hit an E above high C on a regular basis that is excellent, I would suggest you keep practicing the long notes at the bottom of the stave and build up your embouchure.
Tighten the diaphragm as you ascend up into the higher octave, this is the big muscle under your rib cage at the front around the abdomen area.
Always try to produce a nice sound with no blasting, no hard edges etc.
Listen to Maurice Andre playing the Telemann Concerto in D (Clarino) or the Michael Haydn trumpet Concerto where he hits a G above upper C on a D trumpet with virtually no effort at all.
Some of Maynard's best playing is on the newly released Ballad Style of Maynard Ferguson. This album hails from the late 1960's with just a solo trumpet and orchestra, again sounding very effortless!
I hope my little insight is of use and good luck for the future, practice makes perfect as the saying goes!