Post-self-consistent dispersion corrections are among the most popular approaches to incorporate London dispersion into density-functional theory (DFT). Many such corrections have been proposed in the literature, including Grimme's Dn family, the Tkatchenko-Scheffler (TS) model, and the many-body dispersion (MBD) method. The exchange-hole dipole moment (XDM) model is an accurate and non-empirical way of incorporating dispersion that uses a damped asymptotic dispersion energy expression, but calculates the dispersion coefficients to any order from the converged electron density and kinetic-energy density. XDM has been show to be an excellent choice for modelling intermolecular interactions between gas-phase organic molecules, in molecular crystals, layered materials, and more. In this work, we present our recent implementation of XDM in the Fritz Haber Institute ab initio molecular simulations (FHI-aims) electronic structure package, with which we can carry out XDM-corrected DFT calculations using local orbitals. This is an important development because it allows the inexpensive application of XDM-corrected hybrid functionals to periodic solids. The performance of the new implementation was assessed for a battery of tests, including the S22\(\times\)5 and S66\(\times\)8 molecular benchmarks, the X23 set of molecular-crystal lattice energies, and a collection of layered materials. Hybrid functionals based on B86bPBE, in combination with XDM, perform excellently for both the gas-phase and solid-state benchmark tests. In particular, the 50% global hybrid B86bPBE50-XDM achieves a mean absolute error (MAE) for the X23 set of 0.65 kcal/mol, which is the lowest MAE yet reported for any DFT-based method to our knowledge. At the same time, the asymptotic linear scaling of the underlying method makes the new XDM-corrected hybrid functionals an excellent choice for modeling molecular systems, materials, and their interfaces.