We present first evidence for the process e+e-→γηc(1S) at six center-of-mass energies between 4.01 and 4.60 GeV using data collected by the BESIII experiment operating at BEPCII. We measure the Born cross section at each energy using a combination of twelve ηc(1S) decay channels. 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 σpeak(e+e-→γηc(1S))=2.11±0.49(stat.)±0.36(syst.) pb with a statistical significance of 4.2σ.