Spin depolarization of a ZnCdSe quantum-well spin detector (SD) in ZnMnSe/ZnCdSe light-emitting quantum structures is investigated by cw and time-resolved optical orientation spectroscopy. It is shown that spin depolarization is governed by three distinct spin relaxation processes with the corresponding polarization decay times of 850, 30, and <10 ps. The dominant and the fastest process is attributed to spin relaxation accompanying energy relaxation of hot excitons (and hot carriers) within the SD, providing evidence that it can be an important source of spin loss, leading to the observed limited efficiency of optical spin injection in the structures.