Based on an integrated luminosity of $20.3{fb}^{-1}$ of $e^{+}{e}^{-}$ annihilation data collected at a center-of-mass energy of $3.773~\rm{GeV}$ with the BESIII detector operating at the BEPCII collider, an improved search for the radiative transitions $\psi (3770)\to \gamma {\eta }_c(1S,2S)$ is performed using the hadronic decays ${\eta }_c(1S,2S)\to K_S^0{K}^{\pm }{\pi }^{\mp }$. No significant signal is observed. The corresponding 90$\%$ confidence level upper limits on the product branching fractions are set to be $5.0\times {10}^{-6}$ for the ${\eta }_c(1S)$ transition and $3.7\times {10}^{-6}$ for the ${\eta }_c(2S)$ transition. The 90$\%$ confidence level upper limits on the partial decay widths are also reported to be $\Gamma (\psi (3770)\to \gamma {\eta }_c(1S))<5.5$ keV and $\Gamma(\psi(3770) \to \gamma \eta_{c}(2S)) < 29.4~\rm{keV}$. With about seven times larger integrated luminosity than used previously, these results lower the upper limits by approximately a factor of three and two for the ${\eta }_c(1S)$ and ${\eta }_c(2S)$ transitions, respectively.