More lips in the mouthpiece

You can improve your high register a lot by closing you mouth tighter and bringing more chops in the mouthpiece. This will also improve your tone. 

To achieve this, play 3 soft long tones on a note of your choice in the middle-high register. Let's say a f# on top of the staff. Play 16 beats ppp and repeat twice. If the note stops, try to bring the sound back by adding a bit more air and keep playing. It's not the most comfortable exercise but it's simple and very efficient. 

You should immediately feel a difference in your high and soft playing after this.