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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 $δ\sim 8\times 10^{-3}$ 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$^{-1}$.