As the sensitivities of dark matter direct detection experiments improve, they will soon encounter a new source of background, due to coherent elastic neutrino-nucleus scattering (CEvNS). In this talk, I will discuss how the CEvNS rate could vary in certain simplified models of new physics, raising the level of the so-called neutrino floor. I will also discuss prospects for direct detection experiments to put competitive limits on such models, in an energy range beyond the reach of most dedicated neutrino experiments.