After the first observation of coherent elastic neutrino-nucleus scattering (CE$\nu$NS) by the COHERENT collaboration, many efforts are being made to improve the measurement of this process, making it possible to constrain new physics in the neutrino sector. In this paper, we study the sensitivity to non-standard interactions (NSIs) and generalized neutrino interactions (GNIs) of a NaI detector with characteristics similar to the one that is currently being deployed at the Spallation Neutron Source at Oak Ridge National Laboratory. We show that such a detector, whose target nuclei have significantly different proton to neutron ratios (at variance with the current CsI detector), could help to partially break the parameter degeneracies arising from the interference between the Standard Model and NSI contributions to the CE$\nu$NS cross section, as well as between different NSI parameters. By contrast, only a slight improvement over the current CsI constraints is expected for parameters that do not interfere with the SM contribution. We find that a significant reduction of the background level would make the NaI detector considered in this paper very efficient at breaking degeneracies among NSI parameters.
Comment: 31 pages, 10 pdf figures, and 4 tables