  Current long-baseline neutrino-oscillation experiments such as NO$\nu$A and
T2K are mainly sensitive to physics in the neighbourhood of the first
oscillation maximum of the $\nu_\mu \to \nu_e$ oscillation probability. The
future Deep Underground Neutrino Experiment (DUNE) utilizes a wide-band beam
tune optimized for CP violation sensitivity that fully covers the region of the
first maxima and part of the second. In the present study, we elucidate the
role of second oscillation maximum in addressing issues pertaining to unknowns
in the standard three flavour paradigm. We consider a new DUNE beam tune
optimized for coverage of the region of the second oscillation maxima which
could be realized using proposed accelerator upgrades that provide multi-MW of
power at proton energies of 8 GeV. We find that addition of the multi-MW 8 GeV
beam to DUNE wide-band running leads to modest improvement in sensitivity to CP
violation, mass hierarchy, the octant of $\theta_{23}$ as well as the
resolution of $\delta$ and the Jarlskog invariant. Significant improvements to
the DUNE neutrino energy resolution yield a much larger improvement in
performance. We conclude that the standard DUNE wide-band beam when coupled
with excellent detector resolution capabilities is sufficient to resolve
$\delta$ to better than $\sim 12^\circ$ for all values of $\delta$ in a decade
of running. For second maxima (8 GeV 3MW) beam running concurrently with the
standard wide-band (80 GeV 2.2 MW) beam for 5 of the 10 years, it is found that
$\delta$ can be further resolved better than $\sim 10^\circ$ for all values of
$\delta$.
