Coral reefs are widely recognised for providing a natural breakwater effect that modulates erosion and flooding hazards on low‐lying sedimentary reef islands. Increased water depth across reef platforms due sea‐level rise (SLR) can compromise this breakwater effect and enhance island exposure to these hazards, but reef accretion in response to SLR may positively contribute to island resilience. Morphodynamic studies suggest that reef islands can adjust to SLR by maintaining freeboard (island crest elevation above still water level) through overwash deposition and island accretion, but the impact of different future reef accretion trajectories on the morphological response of islands remain unknown. Here we show, using a process‐based morphodynamic model, that, although reef growth significantly affects wave transformation processes and island morphology, it does not lead to decreased coastal flooding and island inundation. According to the model, reef islands evolve during SLR by attuning their elevation to the maximum wave runup and islands fronted by a growing reef platform attain lower elevations than those without reef growth, but have similar overwash regimes. The mean overwash discharge Q_over across the island crest plays a key role in the ability of islands to keep up with SLR and maintain freeboard, with a Q_over value of O(10 l m^‐1 s^‐1) separating island construction from destruction. Islands, therefore, can grow vertically to keep up with SLR via flooding and overwash if specific forcing and sediment supply conditions are met, offering hope for uninhabited and sparely populated islands. However, this physical island response will negatively impact infrastructure and assets on developed islands.