Rates of sea-level rise are increasing across the global ocean. Since $\sim 2008$, sea-level acceleration is particularly pronounced along the US Gulf of Mexico coastline. Here we use model solutions and observational data to identify the physical mechanisms responsible for enhanced rates of recent coastal sea-level rise in this region. Specifically, we quantify the effect of offshore subsurface ocean warming on coastal sea-level rise and its relationship to regional hypsometry. Using the Estimating the Circulation and Climate of the Ocean (ECCO) Version 5 ocean state estimate, we establish that coastal sea-level changes are largely the result of changes in regional ocean mass, reflected in ocean bottom pressure, on interannual to decadal timescales. These coastal ocean bottom pressure changes reflect both net mass flux into and out of the Gulf, as well as internal mass redistribution within the Gulf, which can be understood as an isostatic ocean response to subsurface offshore warming. We test the relationships among coastal sea-level, ocean bottom pressure, and subsurface ocean warming predicted by the model using data from satellite gravimetry, satellite altimetery, tide gauges, and Argo floats. Our estimates of mass redistribution explain a significant fraction of coastal sea-level trends observed by tide gauges. For instance, at St. Petersburg, Florida, this mass redistribution accounts for $>$ 50\% of the coastal sea-level trend observed over the 2008-2017 decade. This study elucidates a physical mechanism whereby coastal sea-level responds to open-ocean subsurface warming and motivates future studies of this linkage in other regions.