Host-range shifts in influenza virus are a major risk factor for pandemics. A key question in the study of emerging zoonoses is how the evolution of transmission efficiency interacts with heterogeneity in contact patterns in the new host species, as this interplay influences disease dynamics and prospects for control. Here we use a synergistic mixture of models and data to tease apart the evolutionary and demographic processes controlling a host-range shift in equine H3N8-derived canine influenza virus (CIV). CIV has experienced 15 years of continuous transfer among dogs in the United States, but maintains a patchy distribution, characterized by sporadic short-lived outbreaks coupled with endemic hotspots in large animal shelters. We show that CIV has a high reproductive potential in these facilities (mean R0 = 3.9) and that these hotspots act as refugia from the sparsely connected majority of the dog population. Intriguingly, CIV has evolved a transmission efficiency that closely matches the minimum required to persist in these refugia, leaving it poised on the extinction/invasion threshold of the host contact network. Corresponding phylogenetic analyses show strong geographic clustering in three US regions, and that the effective reproductive number of the virus (Re) in the general dog population is close to 1.0. Our results highlight the critical role of host contact structure in CIV dynamics, and show how host contact networks could shape the evolution of pathogen transmission efficiency. Importantly, efficient control measures could eradicate the virus, in turn minimizing the risk of future sustained transmission among companion dogs that could represent a potential new axis to the human-animal interface for influenza.