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Abstract The flux spectrum, event rate, and experimental sensitivity are investigated for the diffuse supernova (SN) neutrino background (DSNB), which originates from past stellar collapses and is also known as a supernova relic neutrino background. For this purpose, the contribution of collapses that lead to successful supernova explosion and black hole (BH) formation simultaneously, which are suggested to be a nonnegligible population from the perspective of Galactic chemical evolution, is taken into account. If the BH-forming SNe involve matter fallback onto the protoneutron star for the long term, their total emitted neutrino energy becomes much larger than that of ordinary SNe and failed SNe (BH formation without explosion). Then, in the case of the normal mass hierarchy in neutrino oscillations and with half of all core-collapse SNe being BH-forming SNe, the expected event rate according to the current DSNB model is enhanced by up to a factor of 2 due to the BH-forming SNe. While substantial uncertainties exist regarding the duration of the matter fallback, which determines the total amount of emitted neutrinos, and the fraction of BH-forming SNe, the operation time required to detect the DSNB at Hyper-Kamiokande would be reduced by such contribution in any case.