Using overtones to improve tone and response is tried and true old news. Playing the whole spectrum of overtones improves richness of tone and ease of playing throughout the horn including the lower register, which is great; however, it doesn't specifically address the low register and isn't likely to address all the pertinent details (embouchure and air support) of the low register. That leaves the burgeoning young saxophonist, who wants something specific to do in the low register of the horn, little recourse besides typical long tones, maybe trying different dynamics, vibrato, or zeroing in on intonation. While all of those things are important, they don't efficiently address voicing or embouchure, which are best affected in tandem and fundamental to getting the tone and response you want. What follows is an exercise that specifically addresses voicing in the low register.
Low B-flat: The Gold Standard
What does great a good approach to voicing, embouchure, and air support sound and feel like in the low register? You'll most likely find the answer to this question on low B-flat. It requires healthy air support out of the gate, and due to acoustical properties, most likely the curve of the bow, it generally feels less resistant than its nearby neighbors when playing with a full tone (vs. subtone). Increased air support and it's relative ease in compared to other low notes make it easy to achieve a relaxed embouchure (unless your setup is overly resistant, i.e. reed, mouthpiece) and a huge vibrant sound. This makes low B-flat a logical starting point and a kind of standard by which to judge the rest of our notes.
Try comparing low B-flat to it's closest neighbor, low B, and you'll notice a difference in timbre and in how it feels to play the note. Generally, low B-flat is more vibrant and feels a bit less resistant. As you play B try to maintain the form your embouchure takes as you play low B-flat. Don't let the sides of the bottom lip come in or up. This will go a long way to improving your tone quality and the ease of playing in the low register.
Using the Overtone Above
The next part of the exercise deals more specifically with voicing. For any note that you are looking to improve, you can always improve your voicing by playing the overtone above it and slurring back down into the note. You'll arrive back to the original pitch with an improved and more ideal tongue position, which in turn puts less stress on your embouchure.
First try this on low B-flat. Once you've achieved a beautiful and vibrant B-flat move, up to B and go through the same process. Again focus on maintaining the more relaxed and correct embouchure position you achieve on low B-flat. After improving your low B, try going back and forth between low B-flat and B and see if you can maintain the same ease of playing and rich sound. If you can your on the right track, and you should continue forward.
I find I get the most out of this exercise when I play each note sufficiently long but getting all three notes in one breath and slurring between the notes as shown below.
Continue this process chromatically upward. Be sure to check in with low B-flat often and use it to gauge other notes. Check for rich timbre, embouchure pressure, and easy response.