  We study the supersymmetric extension of $SO(10)$-inspired thermal
leptogenesis showing the constraints on neutrino parameters and on the reheat
temperature $T_{\rm RH}$ that derive from the condition of successful
leptogenesis from next-to-lightest right handed (RH) neutrinos ($N_2$) decays
and the more stringent ones when independence of the initial conditions (strong
thermal leptogenesis) is superimposed. In the latter case, the increase of the
lightest right-handed neutrino ($N_1$) decay parameters helps the wash-out of a
pre-existing asymmetry and constraints relax compared to the non-supersymmetric
case. We find significant changes especially in the case of large $\tan\beta$
values $(\gtrsim 15)$. In particular, for normal ordering, the atmospheric
mixing angle can now be also maximal. The lightest (ordinary) neutrino mass is
still constrained within the range $10 \lesssim m_1/{\rm meV} \lesssim 30$
(corresponding to $75\lesssim \sum_i m_i/{\rm meV} \lesssim 120$). Inverted
ordering is still disfavoured, but an allowed region satisfying strong thermal
leptogenesis opens up at large $\tan\beta$ values. We also study in detail the
lower bound on $T_{\rm RH}$ finding $T_{\rm RH}\gtrsim 1 \times 10^{10}\,{\rm
GeV}$ independently of the initial $N_2$ abundance. Finally, we propose a new
$N_2$-dominated scenario where the $N_1$ mass is lower than the sphaleron
freeze-out temperature. In this case there is no $N_1$ wash-out and we find
$T_{\rm RH} \gtrsim 1\times 10^{9}\,{\rm GeV}$. These results indicate that
$SO(10)$-inspired thermal leptogenesis can be made compatible with the upper
bound from the gravitino problem, an important result in light of the role
often played by supersymmetry in the quest of a realistic model of fermion
masses.
