We study ηc production at center-of-mass energies s from 4.18 to 4.60 GeV in e+e- annihilation data collected with the BESIII detector operating at the BEPCII storage ring, corresponding to 7.3 fb-1 of integrated luminosity. We measure the cross sections of the three different exclusive reactions e+e-→ηcπ+π-π0, e+e-→ηcπ+π-, and e+e-→ηcπ0γ. We find significant ηc production in e+e-→ηcπ+π-π0 at s of 4.23 GeV and 4.26 GeV and observe a significant energy-dependent Born cross section that we measure to be consistent with the production via the intermediate Y(4260) resonance. In addition, we perform a search for a charmoniumlike Zc state close to the DD¯ threshold that decays to ηcπ, involving ground state charmonium, and observe no signal. Corresponding upper limits on the cross section of ηc and Zc production are provided, where the yields are not found to be significant.