  We present first evidence for the process $e^+e^-\to \gamma\eta_c(1S)$ at six
center-of-mass energies between 4.01 and 4.60~GeV using data collected by the
BESIII experiment operating at BEPCII. These data sets correspond to a total
integrated luminosity of 4.6~fb$^{-1}$. We measure the Born cross section at
each energy using a combination of twelve $\eta_c(1S)$ decay channels. Because
the significance of the signal is marginal at each energy ($\le3.0\sigma$), we
also combine all six energies under various assumptions for the
energy-dependence of the cross section. If the process is assumed to proceed
via the $Y(4260)$, we measure a peak Born cross section $\sigma_{\mathrm{
peak}}(e^+e^-\to\gamma\eta_c(1S)) = 2.11 \pm 0.49 (\mathrm{stat.}) \pm 0.36
(\mathrm{syst.})$~pb with a statistical significance of 4.2$\sigma$.
