The spread of SARS-CoV-2 has been geographically uneven. To understand the drivers of this spatial variation in SARS-CoV-2 transmission, in particular the role of stochasticity, we used the early stages of the SARS-CoV-2 invasion in Washington state as a case study. We analysed spatially-resolved COVID-19 epidemiological data using two distinct statistical analyses. The first analysis involved using hierarchical clustering on the matrix of correlations between county-level case report time series to identify geographical patterns in the spread of SARS-CoV-2 across the state. In the second analysis, we used a stochastic transmission model to perform likelihood-based inference on hospitalised cases from five counties in the Puget Sound region. Our clustering analysis identifies five distinct clusters and clear spatial patterning. Four of the clusters correspond to different geographical regions, with the final cluster spanning the state. Our inferential analysis suggests that a high degree of connectivity across the region is necessary for the model to explain the rapid inter-county spread observed early in the pandemic. In addition, our approach allows us to quantify the impact of stochastic events in determining the subsequent epidemic. We find that atypically rapid transmission during January and February 2020 is necessary to explain the observed epidemic trajectories in King and Snohomish counties, demonstrating a persisting impact of stochastic events. Our results highlight the limited utility of epidemiological measures calculated over broad spatial scales. Furthermore, our results make clear the challenges with predicting epidemic spread within spatially extensive metropolitan areas, and indicate the need for high-resolution mobility and epidemiological data.