Spreading broadly refers to the notion of an entity propagating throughout a networked system via its interacting components. Evidence of its ubiquity and severity can be seen in a range of phenomena, from disease epidemics to financial systemic risk. In order to understand the dynamics of these critical phenomena, computational models map the probability of propagation as a function of direct exposure, typically in the form of pairwise interactions between components. By doing so, the important role of indirect interactions remains unexplored. In response, we develop a simple model that accounts for the effect of both direct and subsequent exposure, which we deploy in the novel context of failure propagation within a real-world engineering project. We show that subsequent exposure has a significant effect in key aspects, including the: (a) final spreading event size, (b) propagation rate, and (c) spreading event structure. In addition, we demonstrate the existence of hidden influentials in large-scale spreading events, and evaluate the role of direct and subsequent exposure in their emergence. Given the evidence of the importance of subsequent exposure, our findings offer new insight on particular aspects that need to be included when modelling network dynamics in general, and spreading processes specifically.