A new effect of self-consistency in the relativistic Hartree-Fock (HF) approximation suggested previously is con firmed by a renormalized calculation with the on-shell renormalization conditions. Two self-consistency schemes, ne requiring self-consistency in the HF potential (scheme P) and the other in the baryon propagator (scheme BP), are studied. It is pointed out that the on-shell renormalization conditions make the self-consistency require- ment in scheme P automatically satisfied. Our calculated results show that scheme P is a good approximation to scheme BP for the calculation of the baryon propagator and the self-consistency in scheme BP diminishes the continuum part of the spectral representation for the baryon propagator, while scheme P yields a baryon propagator which is the same as the HF result contributed by the single particle part of the above spectral representation alone. Further, it is demonstrated that the region of validity of the quasi-particle approximationmay depend on the renormalization conditions.