  The full physics potential of the next-generation Deep Underground Neutrino
Experiment (DUNE) is still being explored. In particular, there have been some
recent studies on the possibility of improving DUNE's neutrino energy
reconstruction. The main motivation is that a better determination of the
neutrino energy in an event-by-event basis will translate into an improved
measurement of the Dirac $CP$ phase and other neutrino oscillation parameters.
To further motivate studies and improvements on the neutrino energy
reconstruction, we evaluate the impact of energy resolution at DUNE on an
illustrative new physics scenario, viz. non-standard interactions (NSI) of
neutrinos with matter. We show that a better energy resolution in comparison to
the ones given in the DUNE conceptual and technical design reports may
significantly enhance the experimental sensitivity to NSI, particularly when
degeneracies are present. While a better reconstruction of the first
oscillation peak helps disentangling standard $CP$ effects from those coming
from NSIs, we find that the second oscillation peak also plays a nontrivial
role in improving DUNE's sensitivity.
