Animal populations act as reservoirs for emerging diseases. In order for transmission to be self-sustaining, a pathogen must have a basic reproduction number R0>1. Following a founding transmission event from an animal reservoir to humans, a pathogen has not yet adapted to its new environment and is likely to have an R0<1. However, subsequent evolution may rescue the pathogen from extinction in its new host. Recent applications of branching process theory investigate how the emergence of a novel pathogen is influenced by the number and rates of intermediate evolutionary steps. In addition, repeated contacts between human and reservoir populations may promote pathogen emergence. This article extends a stepping-stone model of pathogen evolution to include reservoir interactions. We demonstrate that the probability of a founding event culminating in an emerged pathogen can be significantly influenced by ongoing reservoir interactions. While infrequent reservoir interactions do not change the probability of disease emergence, moderately frequent interactions can promote emergence by facilitating adaptation to humans. Frequent reservoir interactions promote emergence even with minimal adaptation to humans. Thus, these results warn against perpetuated interaction between humans and animal reservoirs, as occurs when there are ecological or environmental changes that bring humans into more frequent contact with animal reservoirs.