In the past, I was in the habit of never using the octave key for overtones. However, at a lesson with woodwind wizard, Charles Pillow, we were messing around with some difficult overtone exercises, and he was instinctively using the octave key. After some investigation I found that the octave key made overtones, in many respects, easier to play. Even more importantly, I found that using the octave key promoted better technique both in terms of embouchure and voicing.
When To Use The Octave Key
You should use the octave key on every overtone you play, with the exception of the first overtone in every series. For example, in overtone series below, you would not use the octave key on the fundamental pitch, low B-flat, or on the first overtone, middle line B-flat. You would use the octave key on the second overtone (top line F) and all higher overtones.
Benefits of Using the Octave Key
The basic mechanic at play here is that the octave key reduces the resistance or back pressure you feel when playing overtones. This allows you to more easily retain good embouchure technique and not over tighten your embouchure, a common pitfall of practicing strenuous techniques. It also allows you to more easily adjust your vocal tract (tongue, vocal chords, etc.) in shaping the now less resistant air column, which, in turn, allows for more control for less effort. This simple use of the octave key will result in much more efficient and beneficial overtone practice.
You may find playing when playing overtones with the octave key it is more difficult to reach or maintain the correct partial. This is most likely because you have become too dependent on embouchure pressure to perform overtones. Though it will seem like backtracking, learning to play overtones with the octave key will improve your sound and technique.
A quick experiment will illustrate the principle. Play the second overtone of the B-flat series notated above (the second overtone is top line F while holding the fingering for low B-flat). Hold it out as a long tone without the octave key. 15 seconds should do the trick. Now play some music: scales, a melody, or an improvisation. Pay special attention to the tone quality, especially the clarity and impact of each new note when played under a slur.
Now, repeat the long overtone, this time holding the octave key. Play some music again, and listen for the tone. You will notice an increased clarity of tone, a slightly lowered resistance, and each slurred note will have more impact or "pop". These are the results of less overall embouchure pressure and better voicing technique.
If you want to, you can now return to the long overtone without the octave key. You will find that just a short amount of overtone practice can calibrate or alter your technique, and you can shift back and forth between the two settings playing long overtones with and without the octave key.
I would suggest using the octave key as described above for all of your overtone practice. I don't believe that it is a better work out to do overtone practice without it, and instead is, simply, the wrong kind of work out. You will find with the octave key, your practice is more efficient and more consistent.
My new book, which is completely dedicated to overtones, should be out in about a week or so. It's designed to address overtone playing at all levels, so it has exercises to help beginners play their first overtones, but it also has extremely advanced sections that will benefit players who can already play four octaves worth of overtones. I'll put up an official release post soon...