  A new measurement of the quenching factor for low-energy nuclear recoils in
CsI[Na] is presented. Past measurements are revisited, identifying and
correcting several systematic effects. The resulting global data are
well-described by a physics-based model for the generation of scintillation by
ions in this material, in agreement with phenomenological considerations. The
uncertainty in the new model is reduced by a factor of four with respect to an
energy-independent quenching factor initially adopted as a compromise by the
COHERENT collaboration. A significantly improved agreement with Standard Model
predictions for the first measurement of CE$\nu$NS is generated. We emphasize
the critical impact of the quenching factor on the search for new physics via
CE$\nu$NS experiments.
