We study hybrid-molecule structures consisting of a metal nanoparticle (MNP) nanorod coupled to a semiconductor quantum dot (SQD). MNP geometry can be used to tailor the local fields that determine the SQD-MNP coupling and to engineer the hybrid dynamical response. We identify regimes where dark modes and higher-order multipolar modes can influence hybrid response. Strong local-field coupling via dark modes changes the interference and self-interaction effects dramatically. External fields do not directly drive this MNP dark mode, so SQD-MNP coupling is dominated by the local induced self-coupling. By utilizing more complex structures that provide substantial local-field enhancement, we show the strong SQD-MNP coupling regimes can be made much more accessible. All of these aspects could enhance the capabilities of metal nanostructures to provide spatial and spectral control of the optical properties of single emitters.