Using 145 early- and late-type galaxies (ETGs and LTGs) with directly measured supermassive black hole masses, MBH, we build upon our previous discoveries that: (i) LTGs, most of which have been alleged to contain a pseudobulge, follow the relation ; and (ii) the ETG relation is an artifact of ETGs with/without disks following parallel relations that are offset by an order of magnitude in the MBH direction. Here, we searched for substructure in the diagram of MBH versus central velocity dispersion σ, using our recently published multi-component galaxy decompositions, by investigating divisions based on the presence of a depleted stellar core (major dry merger), a disk (minor wet/dry merger, gas accretion), or a bar (evolved unstable disk). The Sérsic and core-Sérsic galaxies define two distinct relations: and , with and 0.46 dex, respectively. We also report on the consistency with the slopes and bends in the galaxy luminosity (L)–σ relation due to Sérsic and core-Sérsic ETGs, and LTGs that all have Sérsic light profiles. Two distinct relations (superficially) reappear in the MBH–σ diagram upon separating galaxies with/without a disk (primarily for the ETG sample), while we find no significant offset between barred and non-barred galaxies, nor between galaxies with/without active galactic nuclei. We also address selection biases purported to affect the scaling relations for dynamically measured MBH samples. Our new MBH–σ relations, dependent on morphological type, more precisely estimate MBH in other galaxies, and hold implications for galaxy/black hole coevolution theories, simulations, feedback, the pursuit of a black-hole fundamental plane, and calibration of virial f-factors for reverberation mapping.