I'm not aware of any real "secret" behind it. Wait. Maybe that's why I'm not a classical clarinet player in a European orchestra making megabucks .
There are a couple different ways of practicing to reach high altissimo, but the general idea is that you want to not only sound "full" but in tune. Get yourself an electronic tuner and play the note in tune -- you might want to ask your instructor to make sure you have a proper embochure, etc., first.
After you can play a note in tune, try another. When you can play that in tune, go back to the first and see if you're still playing in tune. When you can go back and forth and play in tune, add another note. Then speed it up.
One of the solos I practiced on throughout my playing career is a cello solo by Bach, Suite #4 in Eb (while the piece was written for cello, I performed it on Eb bari sax and it can be adapted for any instrument). The main idea behind the first movement is that you have, at the beginning of a bar, a LARGE leap (sometimes two octaves) and then you stay in a high register for the rest of the bar. What you need to master to play the solo properly isn't just good tone, but good intonation -- and that you don't have a "boom" of sound on the first note and then the second note (altissimo) is wimpy. Or worse, you "scoop" the note.
They will instruct different level of players in different ways. I have had to improve my playing over the years to improve my skills especially in altissimo to work on their instruments to their likings.
But a symphony players definition may be different from a college major, to a community band, et all. Thus some of the things below may or may not make much sense to one ears unless you know what you are trying to attain.
 Your embouchure has alot to do with you tone. If you pinch the mpc reed combo you will sound thin to discerning ears.
here's an excerpt
As to how it affects the tone. Start out by using as little mouthpiece as possible. Listen carefully to the tone. Then add more mouthpiece, and more and compare the tone of each location. Mouthpieces were design with certain attributes and we need to play the mouthpiece in the way it was designed
 If you use a soft reed your tone will be affected too. But don't just jump up to a harder reed just to try. One has to work up to a harder reed and practice. Some players jump to a harder reed but end up pinching it to play it immediately, thus creating another problem.
Some instruments are not created equal. before I bought my first pro clarinet I always wondered what is the difference between a pro instrument and my Normandy 4 or Noblet 45 ? At that time I couldn't hear a difference. I would play my N4, N45 and a pro clarinet and hear no real discernable difference. That is until I was taught better embouchure and improved my ear training. Then I could here a difference from top to bottom. But we are talking minute here.
But the altissimmo was more full and clearer on the pro instruments back then compared to the intermediates. You can't really describe this. You either hear it from your expertise or you don't. So you'll either agree or not.
But also the instrument setup may affect it too.
The best advice here is to get with a pro player, preferably a symphony player and take a few lessons from them to see what they think.