Recently, a P,T-odd Faraday effect (PTFE), optical rotation in an external electric field parallel to the light propagation direction, was discussed as a promising tool for observation of the electron electric dipole moment in the intracavity absorption spectroscopy with atoms or molecules. The main mechanism leading to this effect, the linear Stark splitting of atomic levels (PTFE, LS) is well known. This mechanism is similar to the Zeeman splitting mechanism in case of the ordinary Faraday effect, i.e., optical rotation in an external magnetic field. In this paper we analyze the other possible mechanisms of PTFE, in particular, the atomic-level-mixing mechanism (PTFE, LM). In our analysis we perform considering PTFE (as well as the ordinary FE) as a kind of electro- (magneto-) optical circular birefringence. For the ordinary FE the level-mixing mechanism (FE, LM) is known apart from the main Zeeman level splitting mechanism (FE, LS). However, (PTFE, LM) mechanism for the PTFE was never considered in literature. Usually, the (FE, LS) mechanism dominates and the same was expected for (PTFE, LS). We demonstrate, however, that with heavy diatomic molecules, in particular in PbF, the contribution of (PTFE, LM) may essentially exceed the (PTFE, LS). This promises an additional enhancement of P,T-odd effects in experiments with PbF molecule and makes this molecule one of the best candidates for performing such experiments.