During infectious disease epidemics, an important question is whether cases travelling to new locations will trigger local outbreaks. The risk of this occurring depends on the transmissibility of the pathogen, the susceptibility of the host population and, crucially, the effectiveness of surveillance in detecting cases and preventing onward spread. For many pathogens, transmission from pre-symptomatic and/or asymptomatic (together referred to as non-symptomatic) infectious hosts can occur, making effective surveillance challenging. Here, by using SARS-CoV-2 as a case study, we show how the risk of local outbreaks can be assessed when non-symptomatic transmission can occur. We construct a branching process model that includes non-symptomatic transmission and explore the effects of interventions targeting non-symptomatic or symptomatic hosts when surveillance resources are limited. We consider whether the greatest reductions in local outbreak risks are achieved by increasing surveillance and control targeting non-symptomatic or symptomatic cases, or a combination of both. We find that seeking to increase surveillance of symptomatic hosts alone is typically not the optimal strategy for reducing outbreak risks. Adopting a strategy that combines an enhancement of surveillance of symptomatic cases with efforts to find and isolate non-symptomatic infected hosts leads to the largest reduction in the probability that imported cases will initiate a local outbreak.