Suppose that G is an infinite, locally finite connected graph, and { e₁, e₂, … } is a countable dense subset of its space of ends. We prove that G³, the graph obtained from G by adding an edge between any two vertices of distance at most 3 from one another, can be partitioned into a family of one-sided infinite paths { p₁, p₂, … } such that p_i converges to e_i for every i. In particular, if G has only countably many ends, then G³ can be partitioned into a family of paths realizing each of its ends exactly once. This generalizes a result of Seward, that every connected graph of bounded degree with two ends is bi-Lipschitz equivalent to a graph with a bi-infinite Hamiltonian path.