  In recent years there have been many proposals for new electron-positron
colliders, such as the Circular Electron-Positron Collider, the International
Linear Collider, and the Future Circular Collider in electron-positron mode.
Much of the motivation for these colliders is precision measurements of the
Higgs boson and searches for new electroweak states. Hence, many of these
studies are focused on energies above the $h\,Z$ threshold. However, there are
proposals to run these colliders at the lower $WW$ threshold and $Z$-pole
energies. In this paper, we propose a new search for Higgs physics accessible
at lower energies: $e^+e^-\rightarrow h\,Z_d$, where $Z_d$ is a new light gauge
boson such as a dark photon or dark-$Z$. Such searches can be conducted at the
$WW$ threshold, i.e. energies below the $h\,Z$ threshold where exotic Higgs
decays can be searched for in earnest. Additionally, due to very good angular
and energy resolution at future electron-positron colliders, these searches
will be sensitive to $Z_d$ masses below 1 GeV, which is lower than the current
direct LHC searches. We will show that at $\sqrt{s}=160$ GeV with 10 ab$^{-1}$,
a search for $e^+e^-\rightarrow h\,Z_d$ is sensitive to $h-Z-Z_d$ couplings of
$\delta\sim 8\times 10^{-3}$ and cross sections of $\sim 1-2$ ab for $Z_d$
masses below 1 GeV. The results are similar at $\sqrt{s}=240$ GeV with 5
ab$^{-1}$.
