  We study associated Higgs production with a photon at electron-positron
colliders, $e^+e^-\to h\gamma$, in various extended Higgs models, such as the
inert doublet model (IDM), the inert triplet model (ITM) and the two Higgs
doublet model (THDM). The cross section in the standard model (SM) is maximal
around $\sqrt{s}=$250 GeV, and we present how and how much the new physics can
enhance or reduce the production rate. We also discuss the correlation with the
$h\to\gamma\gamma$ and $h\to Z\gamma$ decay rates. We find that, with a sizable
coupling to a SM-like Higgs boson, charged scalars can give considerable
contributions to both the production and the decay if their masses are around
100 GeV. Under the theoretical constraints from vacuum stability and
perturbative unitarity as well as the current constraints from the Higgs
measurements at the LHC, the production rate can be enhanced from the SM
prediction at most by a factor of two in the IDM. In the ITM, in addition, we
find a particular parameter region where the $h\gamma$ production significantly
increases by a factor of about six to eight, but the $h\to\gamma\gamma$ decay
still remains as in the SM. In the THDM, possible deviations from the SM
prediction are minor in the viable parameter space.
