I'll try to relate how I play altissimo on tenor (and alto, and soprano) but it's not easy to accurately describe. For me, starting out was just getting things to pop out any way I could figure out how on my own, but with some instructional guidance from books. I found that nobody could really describe it for me in a way that worked, but I'll try. I have worked on, and continue to work on daily, playing from F#3-G4 as an extension to the horn's range -- not as an effect, but in trying to play up there the same as everywhere else on the horn. I can approach doing that, but not really, because it's a different animal for at least a couple of reasons -- first, the fingerings can be relatively clumsy, and second, it's all harmonics, not the fundamental tone.
Someone mentioned Rascher's Top Tones for the Saxophone. That's an excellent reference. Rousseau's Saxophone High Tones is good too/ There are two big takeaways from that book, for me. The things to work on are 1) Closed Tube Harmonics and 2) Overblowing 6ths. There's a lot more than those two things, but those are almost prerequisite to getting started playing altissimo.
Closed Tube Harmonics: play a low Bb. Now play a low Bb with the octave key, sounding Bb2, an octave up. With the same fingering, play F, Ab, Bb, D, F, Ab and Bb in an arpeggio above that. (some would say it should go Bb2 to Bb3, skipping the F2 and Ab2 in between them, but I can get them all, most of the time) There's the rub: how do you get those to come out? It's by modifying your air stream with your oral cavity, soft palette and throat. Think of your horn as a resonant chamber that starts with the reed, making it a closed cone. Think also of your oral cavity, soft palette and throat as forming a resonant chamber BEFORE the reed. Raise the soft palette; change the tongue position; open your throat to different degrees. You might have to 'bite', that's not optimal, but at first you might find it helpful, and just have to work that back out of your chops later. Likewise, sticking your jaw in or out a couple microns might help at first, but again suboptimal, you'll have to work that back out again later. I'm just talking about how to get started here. You should be able to blow closed tube harmonics like this from low Bb to C#. C# is the hardest to get at first.
The important thing is that you have a resonant chamber both in front of the reed and beyond the reed. Modifying aspects of the one in front of the reed is what enables the harmonics to pop out by forcing the reed to vibrate differently - you can feel it. I'll bet I worked on that for at least a couple years before it started working -- I'm a slow learner about some things like that. In my little mind it's a similar challenge to getting a good sound on flute. My flute sound is still not all that, not even close.
Overblowing 6ths: Starting with high D (D3), you can overblow a 6th above that -- a B4 -- with the same techniques as closed tube harmonics, only with slight variations. All the different harmonics require slight variations of your entire oral cavity. Your embouchure, in the end, shouldn't have to change very much to get them to pop out. You should be able to overblow 6ths from high D to high F. I can get them to come out on C#, but that's more difficult.
When they start popping out, they'll be out of tune. Modify your oral cavity to get them in tune, that's very very important. These two things will not only enable you to start to approach playing altissimo, but like playing long tones, will enable you to 'voice' your sound overall with your oral cavity and improve pitch center and focus. It's hard at first, but worth it.
If you can blow closed tube harmonics and overblow 6ths, then references like altissimo fingering charts will start to work for you. Good luck!