The one-dimensional extended t-V model of fermions on a lattice is a model with long-range interactions that exhibits a transition between a Luttinger liquid conducting phase and a Mott insulating phase. It is known that by tailoring the potential energy of the insulating system, one can force a phase transition into another insulating phase. We show how to construct all possible charge density wave phases of the system with low critical densities in the atomic limit. Higher critical densities are investigated by a brute-force analysis of the basis of finite systems.