  We estimate the sensitivity to the top-quark Yukawa coupling $y_t$ at future
$e^+e^-$ colliders. We go beyond the standard approach that focuses on $t\bar t
h$ production and consider final states with a Higgs boson but not top quarks.
The sensitivity to $y_t$ in such processes comes from the coupling of the Higgs
boson to top quarks in loops. Such final states can be produced in significant
numbers at center-of-mass energies that will be accessible by all proposed
$e^+e^-$ colliders. In a simplified theoretical framework to parametrise
deviations from the Standard Model, we find that at FCC-$ee$ and CEPC operating
at $\sqrt{s}=240$ GeV, $y_t$ could potentially be measured with precision
better than $1\%$. For CLIC and ILC the extraction of $y_t$ could be improved
by a factor of about 2 and 7 respectively, compared to its extraction from just
$t\bar t h$ final states.
