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.
I am associate principal horn of the Singapore Symphony Orchestra and the author of the progressive methods. I'm happy to share my experience as a horn player and teacher with you.