Hospital networks, formed by patients visiting multiple hospitals, affect the spread of hospital-associated infections, resulting in differences in risks for hospitals depending on their network position. These networks are increasingly used to inform strategies to prevent and control the spread of hospital-associated pathogens. However, many studies only consider patients that are received directly from the initial hospital, without considering the effect of indirect trajectories through the network. We determine the optimal way to measure the distance between hospitals within the network, by reconstructing the English hospital network based on shared patients in 2014-2015, and simulating the spread of a hospital-associated pathogen between hospitals, taking into consideration that each intermediate hospital conveys a delay in the further spread of the pathogen. While the risk of transferring a hospital-associated pathogen between directly neighbouring hospitals is a direct reflection of the number of shared patients, the distance between two hospitals far-away in the network is determined largely by the number of intermediate hospitals in the network. Because the network is dense, most long distance transmission chains in fact involve only few intermediate steps, spreading along the many weak links. The dense connectivity of hospital networks, together with a strong regional structure, causes hospital-associated pathogens to spread from the initial outbreak in a two-step process: first, the directly surrounding hospitals are affected through the strong connections, second all other hospitals receive introductions through the multitude of weaker links. Although the strong connections matter for local spread, weak links in the network can offer ideal routes for hospital-associated pathogens to travel further faster. This hold important implications for infection prevention and control efforts: if a local outbreak is not controlled in time, colonised patients will appear in other regions, irrespective of the distance to the initial outbreak, making import screening ever more difficult.