  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.
