  As experimental null results increase the pressure on heavy weakly
interacting massive particles (WIMPs) as an explanation of thermal dark matter
(DM), it seems timely to explore previously overlooked regions of the WIMP
parameter space. In this work we extend the minimal gauged
$U(1)_{L_\mu-L_\tau}$ model studied in \cite{Bauer:2018onh} by a light
(MeV-scale) vector-like fermion $\chi$. Taking into account constraints from
cosmology, direct and indirect detection we find that the standard benchmark of
$M_V=3 m_\chi$ for DM coupled to a vector mediator is firmly ruled out for unit
DM charges. However, exploring the near-resonance region $M_V\gtrsim 2 m_\chi$
we find that this model can simultaneously explain the DM relic abundance
$\Omega h^2 =0.12$ and the $(g-2)_\mu$ anomaly. Allowing for small charge
hierarchies of $\lesssim\mathcal{O}(10)$, we identify a second window of
parameter space in the few-GeV region, where $\chi$ can account for the full DM
relic density.
