  The appealing feature of inverse seesaw models is that the Standard Model
(SM) neutrino mass emerges from the exchange of TeV scale singlets with sizable
Yukawa couplings, which can be tested at colliders. However, the tiny Majorana
mass splitting between TeV singlets, introduced to accommodate small neutrino
masses, is left unexplained. Moreover, we argue that these models suffer from a
structural limitation that prevents a successful leptogenesis if one insists on
having unsuppressed Yukawa couplings and TeV scale singlets. In this work we
propose a hybrid seesaw model, where we replace the mass splitting with a
coupling to a high scale seesaw module including a TeV scalar. We show that
this structure achieves the goal of filling both the above gaps with couplings
of order unity. The necessary structure automatically arises embedding the
seesaw mechanism in composite Higgs models, but may also be enforced by new
gauge symmetries in a weakly-coupled theory. Our hybrid seesaw models have
distinguishing features compared to the standard high scale type-I seesaw and
inverse seesaw. Firstly, they have much richer phenomenology. Indeed, they
generally predict new TeV scale physics (including scalars) potentially
accessible at present and future colliders, whereas weakly-coupled versions may
also have astrophysical and cosmological signatures due to the presence of a
light Nambu-Goldstone boson coupled to neutrinos. Secondly, our scenario
features an interesting interplay between high scale and TeV scale physics in
leptogenesis and enlarges the range of allowed high scale singlet masses beyond
the usual $\sim 10^9-10^{ 15 }$ GeV, without large hierarchies in the Yukawa
couplings nor small mass splitting among the singlets.
