We study the consequences of a hadron-quark phase transition (PT) in failing core-collapse supernovae (CCSNe) that give birth to stellar-mass black holes (BH). We perform a suite of neutrino-transport general-relativistic hydrodynamic simulations in spherical symmetry with 21 progenitor models and a hybrid equation of state (EoS) including hadrons and quarks. We find that the effect of the PT on the CCSN postbounce dynamics is a function of the bounce compactness parameter xi(2.2). For xi(2.2) greater than or similar to 0.24, the PT leads to a second dynamical collapse of the protocompact star (PCS). While BH formation starts immediately after this second collapse for models with xi(2.2) greater than or similar to 0.51, the PCS experiences a second bounce and oscillations for models with 0.24 less than or similar to x xi(2.2) less than or similar to 0.51. These models emit potent oscillatory neutrino signals with a period of similar to 1 ms for tens of milliseconds after the second bounce, which can be a strong indicator of the PT in failing CCSNe if detected in the future. However, no shock revival occurs and BH formation inevitably takes place in our spherically symmetric simulations. Furthermore, via a diagram of mass-specific entropy evolution of the PCS, the progenitor dependence can be understood through the appearance of a third family of compact stars emerging at large entropy induced by the PT.