A search has been performed for neutrinos from two sources, the $hep$ reaction in the solar $pp$ fusion chain and the $\nu_e$ component of the diffuse supernova neutrino background (DSNB), using the full dataset of the Sudbury Neutrino Observatory. The $hep$ search is performed using both a single-bin counting analysis and a likelihood fit. We find a best-fit flux that is compatible with solar model predictions while remaining consistent with zero flux, and set a one-sided upper limit of $\Phi_{hep} <30\times10^{3}~\mathrm{cm}^{-2}~\mathrm{s}^{-1}$ (90% credible interval). No events are observed in the DSNB search region, and we also set an improved upper bound on the $\nu_e$ component of the DSNB flux.