  Models with an extended scalar electroweak sector are well motivated. Such
models could accommodate a dark matter candidate if there is an additional
scalar representation with a vanishing vacuum expectation value and, in
addition, there are no couplings between fermions and the dark matter
candidate. The most natural way to have these conditions implemented is to
consider models where an underlying symmetry is imposed. Governed by this, we
consider a three-Higgs-doublet model with an $S_3$ symmetry. Within this
framework there are different implementations which could possibly accommodate
a dark matter candidate. The family of $S_3$-symmetric three-Higgs-doublet
implementations arises due to different vacua and, as a result, different
minimisation conditions. In this framework the dark matter candidate falls into
the class of weakly interacting massive particles. The dark matter candidate is
associated with an $\mathbb{Z}_2$ symmetry which survives spontaneous symmetry
breaking and is a remnant of the $S_3$ symmetry. We explore two cases, they
share many aspects of the Type-I two-Higgs-doublet model plus an inert SU(2)
doublet. The main difference between these two cases is the presence of an
irremovable phase, which leads to CP violation in one of the implementations.
The two candidate cases differ from other previously studied models with three
scalar doublets by the fact that they do not allow for heavy dark matter
candidates, $\mathcal{O}(500)\text{ GeV}$. Valid dark matter regions were
identified as $m_\mathrm{DM} \in [52.5,\,89]~\text{GeV}$ for a model without CP
violation and $m_\mathrm{DM} \in [6.5,\,44.5]~\text{GeV}$ for a model with CP
violation. In the present work we refine the parameter space by applying
additional checks to our previous work coming from LHC data and from indirect
detection data.
