Quantum simulators offer new ways of addressing the long-range entanglement structure of topologically ordered phases of matter. However, measuring entanglement on large subsystems is costly, such that the detection of topological entanglement across long distances becomes unfeasible. Here, we propose a protocol to overcome this challenge by adiabatically deforming the state into geometries where universal long-range entanglement can be efficiently extracted from smaller subregions. Our protocol is general and readily applicable to various quantum simulation architectures. We illustrate our scheme for various string-net models representing both abelian and non-abelian topologically ordered phases, and illustrate an application with neutral atom tweezer arrays.