  The recent measurement of the muon anomalous magnetic moment by the Fermilab
E989 experiment, when combined with the previous result at BNL, has confirmed
the tension with the SM prediction at $4.2\,\sigma$ CL, strengthening the
motivation for new physics in the leptonic sector. Among the different particle
physics models that could account for such an excess, a gauged
$U(1)_{L_\mu-L_{\tau}}$ stands out for its simplicity. In this article, we
explore how the combination of data from different future probes can help
identify the nature of the new physics behind the muon anomalous magnetic
moment. In particular, we contrast $U(1)_{L_\mu-L_{\tau}}$ with an effective
$U(1)_{L_\mu}$-type model. We first show that muon fixed target experiments
(such as NA64$\mu$) will be able to measure the coupling of the hidden photon
to the muon sector in the region compatible with $(g-2)_\mu$, and will have
some sensitivity to the hidden photon's mass. We then study how experiments
looking for coherent elastic neutrino-nucleus scattering (CE$\nu$NS) at
spallation sources will provide crucial additional information on the kinetic
mixing of the hidden photon. When combined with NA64$\mu$ results, the
exclusion limits (or reconstructed regions) of future CE$\nu$NS detectors will
also allow for a better measurement of the mediator mass. Finally, the
observation of nuclear recoils from solar neutrinos in dark matter direct
detection experiments will provide unique information about the coupling of the
hidden photon to the tau sector. The signal expected for
$U(1)_{L_\mu-L_{\tau}}$ is larger than for $U(1)_{L_\mu}$ with the same kinetic
mixing, and future multi-ton liquid xenon proposals (such as DARWIN) have the
potential to confirm the former over the latter.
