  Many models have recently been proposed in which dark matter (DM) couples to
Standard Model fields via a GeV-scale dark sector. We consider scenarios of
this type where the DM mass, at the electroweak/TeV scale, is generated by the
VEV of a singlet which also couples to the Higgs. Such a setup results in a
distinct recoil spectrum with both elastic and inelastic components. We
construct an explicit NMSSM-like realization of this setup, discuss constraints
coming from the relic density, and include benchmark points which are
consistent with current limits, yet visible at upcoming direct detection
experiments.
