  We develop an extension of the basic inverse seesaw model which addresses
simultaneously two of its drawbacks, namely, the lack of explanation of the
tiny Majorana mass term $\mu$ for the TeV-scale singlet fermions and the
difficulty in achieving successful leptogenesis. Firstly, we investigate
systematically leptogenesis within the inverse (and the related linear) seesaw
models and show that a successful scenario requires either small Yukawa
couplings, implying loss of experimental signals, and/or quasi-degeneracy among
singlets mass of different generations, suggesting extra structure must be
invoked. Then we move to the analysis of our new framework, which we refer to
as hybrid seesaw. This combines the TeV degrees of freedom of the inverse
seesaw with those of a high-scale ($M_N\gg$ TeV) seesaw module in such a way as
to retain the main features of both pictures: naturally small neutrino masses,
successful leptogenesis, and accessible experimental signatures. We show how
the required structure can arise from a more fundamental theory with a gauge
symmetry or from warped extra dimensions/composite Higgs. We provide a detailed
derivation of all the analytical formulae necessary to analyze leptogenesis in
this new framework, and discuss the entire gamut of possibilities our scenario
encompasses: including scenarios with singlet masses in the enlarged range $M_N
\sim 10^6 - 10^{16}$ GeV. The idea of hybrid seesaw was proposed by us in
arXiv:1804.06847; here, we substantially elaborate upon and extend earlier
results.
