Abstract Mapping subglacial topography along the margin of the Greenland Ice Sheet was revolutionized by reconciling measurements of ice thickness and surface velocity using the principle of mass conservation. Despite evidence that many subglacial valleys resolved by that method extend upstream, it cannot be applied to the ice sheet’s slower‐flowing interior, where ice thickness observations are sparse. Here we apply Ice Flow Perturbation Analysis to surface elevation and velocity observations, revealing a vast, island‐wide network of subglacial valleys connected to the peripheral valleys resolved by the mass conservation method. The morphology of this valley network further implicates Greenland’s southern and eastern highlands as inception points for the ice sheet and also suggests that this network is modulated by both groundwater flow and Greenland’s tectonic inheritance.