Traditional financial exposure measures primarily focus on direct losses from counterparty default. Following the 2007-2008 financial crisis, the significance of indirect exposures through shared assets gained recognition. However, neither direct nor indirect measures fully account for losses stemming from shock propagation and amplification after a default. This paper introduces "higher-order exposures" to address these spill-over losses and proposes a method for their quantification. Utilizing detailed data from South Africa's banking and investment fund sectors, a contagion model was employed to study common contagion channels and their interactions. The study demonstrates that higher-order exposures constitute a substantial portion of total exposures, especially during financial distress. Furthermore, these higher-order exposures are not predictable from direct or indirect measures alone, as they are heavily influenced by network structure and institutional resilience. The research underscores the need to consider direct, indirect, and higher-order exposures when designing and implementing regulatory tools. These tools include large exposure limits, capital requirement calibration, stress testing, and resolution planning. Ignoring higher-order exposures could lead to inadequate regulation and ineffective crisis management.