Alkali halides are well-known for their tendency to form rock-salt-like crystal structures. Here we present a scanning tunneling microscopy study of a previously unreported alternative structure of one such alkali halide, RbI. When deposited on Ag(111) at a low submonolayer surface coverage, RbI forms islands with hexagonally coordinated atomic structures, in contrast to the expected rock-salt structures typically observed for such alkali halide films on metal surfaces. At a near-monolayer RbI surface coverage, we observe the coexistence of the hexagonally coordinated phase and a square-coordinated rock-salt-like RbI phase that is analogous to that observed for other alkali halides. Our density functional theory calculations for this system highlight the role of RbI-Ag interfacial charge transfer in defining the RbI structure and the impact of local atomic coordination on the RbI-Ag charge-transfer interaction.