Pre-white dwarfs still have double nuclear burning shells. We investigate the nuclear driven mechanism in pre-DA white dwarfs that have substantial hydrogen shell-burning, by carrying out a fully non-adiabatic analysis for oscillations of those stars evolved from the post-asymptotic giant branch. It is shown that nuclear reactions in the hydrogen burning-shell excite low-degree g modes in the period range of about 50-200 s for the pre-white dwarfs with Teff = 40,000 K - 300,000 K. It is also shown that the amount of hydrogen gives a significant influence on the instability domain of such pre-white dwarfs in the Hertzsprung-Russel diagram. Thus, the thickness of hydrogen-dominated envelopes can be well constrained by observaing the presence of the g-mode oscillations. This opens a new window of asteroseismology to unveil the invisible interior of pre-white dwarfs nd the relevant unsolved physics.