  We report the measurement of $e^+e^- \to D^+_sD^*_{s2}(2573)^-+c.c.$ via
initial-state radiation using a data sample of an integrated luminosity of
921.9 fb$^{-1}$ collected with the Belle detector at the $\Upsilon(4S)$ and
nearby. We find evidence for an enhancement with a 3.4$\sigma$ significance in
the invariant mass of $D^+_sD^*_{s2}(2573)^- +c.c.$ The measured mass and width
are $(4619.8^{+8.9}_{-8.0}({\rm stat.})\pm2.3({\rm syst.}))~{\rm MeV}/c^{2}$
and $(47.0^{+31.3}_{-14.8}({\rm stat.})\pm4.6({\rm syst.}))~{\rm MeV}$,
respectively. The mass, width, and quantum numbers of this enhancement are
consistent with the charmonium-like state at 4626 MeV/$c^2$ recently reported
by Belle in $e^+e^-\to D^+_sD_{s1}(2536)^-+c.c.$ The product of the $e^+e^-\to
D^+_sD^*_{s2}(2573)^-+c.c.$ cross section and the branching fraction of
$D^*_{s2}(2573)^-\to{\bar D}^0K^-$ is measured from $D^+_sD^*_{s2}(2573)^-$
threshold to 5.6 GeV.
