The Lieb-Liniger model describes one-dimensional bosons interacting through a repulsive contact potential. In this work, we introduce an extended version of this model by replacing the contact potential with a decaying exponential. Using the recently developed continuous matrix product states techniques, we explore the ground-state phase diagram of this model by examining the superfluid and density correlation functions. At weak coupling superfluidity governs the ground state, in a similar way as in the Lieb-Liniger model. However, at strong coupling quasicrystal and super-Tonks-Girardeau regimes are also found, which are not present in the original Lieb-Liniger case. Therefore the presence of the exponentially decaying potential leads to a superfluid/super-Tonks-Girardeau/quasicrystal crossover, when tuning the coupling strength from weak to strong interactions. This corresponds to a Luttinger liquid parameter in the range K(0,∞), in contrast with the Lieb-Liniger model, where K[1,), and the screened long-range potential, where K(0,1].