  We consider extensions of the next-to-minimal supersymmetric model (NMSSM) in
which the observed neutrino masses are described in terms of effective
dimension six (or seven) rather than dimension five operators. All such
operators respect the discrete symmetries of the model. The new particles
associated with the double (or triple) seesaw mechanism can have sizable
couplings to the known leptons, even with a TeV seesaw scale. In the latter
case some of these new short-lived particles could be produced and detected at
the LHC.
