Density functional theory (DFT)-1/2 is an efficient bandgap rectification method for DFT under local density approximation (LDA) or generalized gradient approximation. It was suggested that non-self-consistent DFT-1/2 should be used for highly ionic insulators like LiF, whereas self-consistent DFT-1/2 should still be used for other compounds. Nevertheless, there is no quantitative criterion prescribed for which implementation should work for an arbitrary insulator, which leads to severe ambiguity in this method. In this work, we analyze the impact of self-consistency in DFT-1/2 and shell DFT-1/2 calculations in insulators or semiconductors with ionic bonds, covalent bonds, and intermediate cases and show that self-consistency is required even for highly ionic insulators for globally better electronic structure details. The self-energy correction renders electrons more localized around the anions in self-consistent LDA-1/2. The well-known delocalization error of LDA is rectified, but with strong overcorrection, due to the presence of additional self-energy potential. However, in non-self-consistent LDA-1/2 calculations, the electron wave functions indicate that such localization is much more severe and beyond a reasonable range because the strong Coulomb repulsion is not counted in the Hamiltonian. Another common drawback of non-self-consistent LDA-1/2 is that the ionicity of the bonding gets substantially enhanced, and the bandgap can be enormously high in mixed ionic-covalent compounds like TiO2.