  With just the Standard Model Higgs doublet, there are only three types of
seesaw models that generate light Majorana neutrino masses at tree level after
electroweak spontaneous symmetry breaking. However, if there exist additional
TeV scalars acquiring vacuum expectation values, coupled with heavier fermionic
multiplets, several new seesaw models become possible. These new seesaws are
the primary focus of this study and correspond to the tree-level ultraviolet
completions of the effective operators studied in a companion publication. We
are interested in the genuine cases, in which the standard seesaw contributions
are absent. In addition to the tree-level generation of neutrino masses, we
also consider the one-loop contributions. Furthermore, we construct low-energy
versions that exhibit a very rich phenomenology. Specifically, we scrutinise
the generation of dimension-6 operators and explore their implications,
including non-unitarity of the leptonic mixing matrix, non-universal $Z-$boson
interactions, and lepton flavor violation. Finally, we provide (Generalised)
Scotogenic-like variants that incorporate viable dark matter candidates.
