  We compute the one-loop corrections to $\sigma(e^+ e^- \to Z h)$ arising from
representative extended Standard Model scalar sector scenarios. According to
the new scalar $\mathrm{SU(2)_L}$ representations, we consider the inert
doublet, real and complex triplet, quintuplet, and septuplet models. With the
sub-percent level precision expected for prospective future $e+e-$ collider
measurements of $\sigma(e^+ e^- \to Z h)$, studies of the Higgsstrahlung
process will probe extended scalar sector particle spectrum and interactions in
a manner complementary to direct searches at the Large Hadron Collider and
possible future $pp$ colliders. We also compare with the sensitivity of future
Higgs diphoton decay rate measurements. We find that the $\sigma(e^+ e^- \to Z
h)$ and $\Gamma(h\to\gamma\gamma)$ complementarity is particularly pronounced
for the complex triplet model.
