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Reactor neutrino experiments have seen major improvements in precision in recent years. With the experimental uncertainties becoming lower than those from theory, carefully considering all sources of $\overlineν_{e}$ is important when making theoretical predictions. One source of $\overlineν_{e}$ that is often neglected arises from the irradiation of the nonfuel materials in reactors. The $\overlineν_{e}$ rates and energies from these sources vary widely based on the reactor type, configuration, and sampling stage during the reactor cycle and have to be carefully considered for each experiment independently. In this article, we present a formalism for selecting the possible $\overlineν_{e}$ sources arising from the neutron captures on reactor and target materials. We apply this formalism to the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory, the $\overlineν_{e}$ source for the the Precision Reactor Oscillation and Spectrum Measurement (PROSPECT) experiment. Overall, we observe that the nonfuel $\overlineν_{e}$ contributions from HFIR to PROSPECT amount to 1\% above the inverse beta decay threshold with a maximum contribution of 9\% in the 1.8--2.0~MeV range. Nonfuel contributions can be particularly high for research reactors like HFIR because of the choice of structural and reflector material in addition to the intentional irradiation of target material for isotope production. We show that typical commercial pressurized water reactors fueled with low-enriched uranium will have significantly smaller nonfuel $\overlineν_{e}$ contribution.