The scramble number of a graph is an invariant recently developed to study chip-firing games and divisorial gonality. In this paper we introduce the screewidth of a graph, based on a variation of the existing literature on tree-cut decompositions. We prove that this invariant serves as an upper bound on scramble number, though they are not always equal. We study properties of screewidth, and present results and conjectures on its connection to divisorial gonality.