Upcoming cosmological surveys will measure the large-scale distribution of galaxies at the subpercent level. In order to extract unbiased cosmological data while retaining valuable small-scale information, we need highly accurate models of the connection between galaxies and (dark) matter. While cosmological hydrodynamical simulations are too small and computationally expensive to directly use in the analysis of galaxy observations, they provide a detailed probe of the galaxy-halo link (under the assumptions of a particular, plausible galaxy formation model). We show that the simplest galaxy-halo model, the mass-only halo occupation distribution (HOD), fails to capture the galaxy clustering at the 15% level, which is well beyond the 1% requirement set by current and future experiments. We develop augmented models which reproduce multiple galaxy distribution statistics by the hydro simulation. Applying these models to observational data can alleviate existing tensions (e.g., Lensing is low) and provide an accurate intermediate-scale prediction, which is critical for upcoming surveys.