Welcome to Gaia! ::

I'd say your number one concern with any note above high C (thumb, register) is intonation first, then you can worry about tone. I've been playing 8 years taking private lessons virtually the whole time and my notes can sound great, but often the pitch is the issue. If you can play the notes in tune with the appropriate dynamic marking, no one will yell at you for tone until you're performing solos in front of large audiences of fellow woodwind players.

The first tip I have for you is to look up fingerings. Each unique fingering has its own distinctive sound, and finding the right one for your instrument will help a lot with both tone and intonation. There are many fingering charts online, although in my experience they tend to give you too many that don't work very well. I'd suggest asking teachers and higher chairs what they use and what alternatives there are. Some instruments just don't sound great in the altissimo register anyways, so you may have to upgrade if that's the case.

The other advice I'll offer you is to NOT PINCH your embouchure. I have fallen into that habit, particularly since I play on strength 5 reeds (only because they play well on my M15 mouthpiece and I use enough air) and I assure you the earlier you learn that, the better. You may need to squeze if you're getting tired during a performance, but be aware that you'll run very sharp, so you'll have to compensate. Try applying more pressure to the sides of your lips rather than biting down on the mouthpiece, then as those muscles develop, it will get easier to get the higher notes to speak and tune, and the sound quality will improve over time as well.

I hope you found this useful, and good luck with those high notes. Remember that there are alternative or trick fingerings for some of the notes that should be used depending on the context of the piece you're playing.