  In the absence of high-statistics supernova neutrino measurements, estimates
of the diffuse supernova neutrino background (DSNB) hinge on the precision of
simulations of core-collapse supernovae. Understanding the cooling phase of
protoneutron star (PNS) evolution ($\gtrsim1\,{\rm s}$ after core bounce) is
crucial, since approximately 50% of the energy liberated by neutrinos is
emitted during the cooling phase. We model the cooling phase with a hybrid
method by combining the neutrino emission predicted by 3D hydrodynamic
simulations with several cooling-phase estimates, including a novel
two-parameter correlation depending on the final baryonic PNS mass and the time
of shock revival. We find that the predicted DSNB event rate at
Super-Kamiokande can vary by a factor of $\sim2-3$ depending on the
cooling-phase treatment. We also find that except for one cooling estimate, the
range in predicted DSNB events is largely driven by the uncertainty in the
neutrino mean energy. With a good understanding of the late-time neutrino
emission, more precise DSNB estimates can be made for the next generation of
DSNB searches.
