Aims. We present a new microscopic hadron-quark hybrid equation of state model for astrophysical applications, from which compact hybrid star configurations are constructed. These are composed of a quark core and a hadronic shell with a first-order phase transition at their interface. The resulting mass-radius relations are in accordance with the latest astrophysical constraints. Methods. The quark matter description is based on a quantum chromodynamics (QCD) motivated chiral approach with higher-order quark interactions in the Dirac scalar and vector coupling channels. For hadronic matter we select a relativistic mean-field equation of state with density-dependent couplings. Since the nucleons are treated in the quasi-particle framework, an excluded volume correction has been included for the nuclear equation of state at suprasaturation density which takes into account the finite size of the nucleons. Results. These novel aspects, excluded volume in the hadronic phase and the higher-order repulsive interactions in the quark phase, lead to a strong first-order phase transition with large latent heat, i.e. the energy-density jump at the phase transition, which fulfils a criterion for a disconnected third-family branch of compact stars in the mass-radius relationship. These twin stars appear at high masses (~2 M_sun) that are relevant for current observations of high-mass pulsars. Conclusions. This analysis offers a unique possibility by radius observations of compact stars to probe the QCD phase diagram at zero temperature and large chemical potential and even to support the existence of a critical point in the QCD phase diagram.