The fingerings above the two and a half octaves found in most beginner lesson books are of course played on overtones. The fingerings aren't "linear" like they are for the lower range, but are no more complex than what many people use for p-whistle or recorder.
For a starting point, finger the low Bb (all pads closed1) and practice playing it like a bugle, without moving your fingers and using only lip control to "overblow" harmonics of the low Bb. With a little practice at this, you should be fairly easily able to blow all the harmonics up to the Bb four octaves above the tonic note of the horn. When you are able to produce all these overtones, you're ready to move from the "just tuned" bugle notes to the fingered chromatic notes in the upper one and a half octaves.
Note that it helps a little to have an actual "saxophone" mouthpiece. The bore of the mouthpiece should continue the conical taper of the instrument itself, and the "flat" that the reed presses against should have an even curve to it so that pressing the reed against it moves the contact point toward the tip. It's control of the "free length" of the reed that allows you to control the overtones. Quite a few "jazzed up" mouthpieces have a deviant sort of mouthpiece bore, and a very flat rim where the reed seats, to intentionally produce a "rougher" tone. Buescher was noted for generally using a "proper" sax mouthpiece when I was playing actively, but most others are "close enough" unless you've got one of the really extreme ones.
The "textbook" I used as a kid is long out of print: Top Tones for Saxophone by Sigurd Rascher. I'll have to do a bit of searching to see if there's another more current one that would be easy to find.
1 The low Bb is the normal bottom note for all saxophones, although I've seen ads for a few that add an A a semitone lower. Just use the whole bore for your bugle practice.