Recently, much attention has been paid to some new carbon-fluoride gases, such as C2F4H2 (R134), C3F4H2 (HFO1234ze), C4F8, C4F7N, C5F10O, and C6F12O. All these gases present excellent dielectric strength and very low values of global warming potential and thus have potential to replace sulfur hexafluoride (SF6) as an insulating medium. However, the electron-impact ionization cross sections (Qion), which are the necessary basic data for the investigation of plasma discharges in these gases, are still unavailable. This work is devoted to the calculation of Qionn for the above new SF6 replacements with incident electron energy of up to 5000 eV. We propose a method of combining the Deutsch-Märk (DM) formalism at low electron energy and the Binary-Encounter-Bethe (BEB) formalism at high electron energy by using a dual sigmoid function. The comparison between the calculated and experimental Qion for various molecules indicates that this new combined method yields 49.41% and 61.24% improvement by the Euclidean metric on average compared with the BEB and DM methods, respectively. We also investigate the effects of molecular orbitals and atomic shells on the Qion of new SF6 replacements. It is found that the BEB method conforms to the law that higher molecular orbitals have larger effect on electron-impact ionization of a molecule, while the DM method does not. Generally, the 2p shells of carbon, nitrogen, or oxygen atoms in a molecule play a dominant role in electron-impact ionization, whereas the shells of fluorine atoms do not affect the ionization process very significantly.