  We explore a possible explanation for the hierarchy in scale between the
atmospheric and solar neutrino mass differences ($\lvert \Delta m^{2}_{31}
\rvert$, and $\Delta m^{2}_{21}$) through the presence of two distinct neutrino
mass mechanisms from tree- and one-loop-level contributions. We demonstrate
that the ingredients needed to explain this hierarchy are present in the
minimal discrete dark matter model [arXiv:2301.10811]. This scenario is
characterized by adding new RH neutrinos and scalar $SU(2)$ doublets to the
Standard Model as triplet representations of an $A_4$ flavour symmetry. The
$A_4$ symmetry breaking, which occurs at the electroweak scale, leads to a
residual $\mathbb{Z}_2$ symmetry responsible for the dark matter stability and
dictates the neutrino phenomenology. We show that CP breaking in the scalar
potential is needed to fit the neutrino mixing angles.
