  We investigate neutrinoless double-beta decay ($0\nu\beta\beta$) in the
minimal extension of the standard model of particle physics, the $\nu$SM, where
gauge-singlet right-handed neutrinos give rise to Dirac and Majorana neutrino
mass terms. We focus on the associated sterile neutrinos and argue that the
usual evaluation of their contributions to $0\nu\beta\beta$, based on
mass-dependent nuclear matrix elements, is missing important contributions from
neutrinos with ultrasoft and hard momenta. We identify the hadronic and nuclear
matrix elements that enter the new contributions, and calculate all relevant
nuclear matrix elements for $^{136}$Xe using the nuclear shell model. Finally,
we illustrate the impact on $0\nu\beta\beta$ rates in specific neutrino mass
models and show that the new contributions significantly alter the
$0\nu\beta\beta$ rate in most parts of the $\nu$SM parameter space.
