Take a high-school male voice student whose range would best be classified as a bass. The top of his range is about F3. However, he can also sing falsetto. The lowest falsetto note for him is about G4. So, there is over an octave he seems unable to access. In certain choirs with few men, that means that much of the repertoire is unsingable for him, since the range of the one "baritone" part often falls entirely in the gap between the main part of his range and the falsetto.
One exercise I have tried is the "siren" exercise, in which he starts a tone at the bottom of his range, sings portamento up to the highest pitch possible for him, then returns portamento to the bottom. The gap is present in this exercise as well.
Are there any exercises that can be done to help him gain at least some of this "missing" range? If so, what exercises?