Using data collected with the BESIII detector operating at the Beijing Electron Positron Collider, we search for the process e+e-→ηcηπ+π-. The search is performed using five large datasets recorded at center-of-mass energies of 4.23, 4.26, 4.36, 4.42, and 4.60 GeV. The ηc meson is reconstructed in 16 exclusive decay modes. No signal is observed in the ηc mass region at any center-of-mass energy. The upper limits on the reaction cross sections are determined to be 6.2, 10.8, 27.6, 22.6 and 23.7 pb at the 90% confidence level at the center-of-mass energies listed above.