The way I was introduced to overtones, and the way I continue to use them, is for voicing. Before I was taught the altissimo register, my instructor spent months working on the overtone series with me. The Sigurd Rascher book Top Tones for the Saxophone was the book that my instructor used with me ATT. In order to successfully play overtones, you of course have to hear the notes in your head before they come out of the horn.
Altissimo playing just takes this to the next level, since now you're combining different fingerings to obtain different notes, as well as the hearing of notes before they come out of the horn. What makes altissimo challenging is that the notes are--or can be--in quick progression, and in some cases may not be within a normal sound range of what the tune calls for. (If for example, your playing outside the changes.) To illustrate this last point, let me give you an example from my own playing experience.
In the blues band Deception I worked in for a decade, we covered the famous Blues Brother tune (I Got Everything I Need) Almost. ATM I can't remember what key we played in, but the tenor solo started on a sustained F#5 and did a glissando down. The problem for me was that F# was not a normal note in the blues key we were playing in. I had a hell of a time trying to hear a F# before it came out of my horn. At rehearsal I blew it more than 50% of the time. It became a running joke in the band. For performances I only f'd it up once--and that was the first time we played it publicly--and the song was a hugely popular one for us, so we played it almost at all performances.
So getting back to your question, the overtone series is a tool that allows you to develop your voicing. When I've been away from altissimo playing for a while, I go back to my overtone series to get my muscle memory in my throat, etc. back. Make sense?
