We present a series of novel numerical results concerning the one-dimensional fermionic Hubbard model with density-dependent hopping amplitude, originally considered in the context of hole superconductivity and recently re-considered for its realizability in a optical lattice setup. In particular we explore the phases of the model as the particle filling and the population imbalance (magnetization) are varied. Focusing on the neighborhood of the regions where the system exhibits a phase separation on mesoscopic scales, we study in detail different types of correlation functions to single out the situations where the superconducting ones dominate.