Soft matter systems can achieve structural variety beyond that of their atomic and molecular building blocks. In past experimental work, soft materials, e.g., nanoparticles and colloids, have been shown to spontaneously self-assemble into familiar crystallographic arrangements, ranging from simple, close-packed structures to some of the most complex known crystal structures such as clathrates. Simulations using coarse-grained molecular dynamics simulations can also yield similar structures, as well as structures with no known atomic equivalents. Particles controlled by relatively simple, short-range isotropic pair potentials have been shown to generate such complex assemblies as a one-component icosahedral quasicrystal---a highly ordered but aperiodic type of structure, which had previously only been observed in multi-component intermetallics. The search for new, self-assembling crystal structures hinges on the exact interaction potential, yet how the shape of the interaction potential influences the resulting structure is still poorly understood. In this work, we design a new functional form for the interaction potential, in which its features can be tuned independently and intuitively. We use our interaction potential to screen small regions of phase space by sampling two-dimensional cuts, targeting regions of low-coordinated crystal structures. We report the discovery and characterization of eight new crystal structures---cI16-X, cP8-X, hP3-X, hP14-Y, hR3-X, mC32-X, tI16-X, and tI32-Y---as well as six more structures that require more advanced characterization techniques.