An unusual fine spectrum of very narrow lines (<1meV overall) is observed in resonance Raman scattering from a CdTe quantum well under normal light incidence, when the laser excitation is about one longitudinal optical (LO) phonon energy higher than the quantum well exciton ground state. The visibility of this fine spectrum is improved by sample doping which reduces the quantum yield of the exciton photoluminescence. The appearance of the four sideway lines of the fine spectrum is explained by the Raman scattering on the combination of LO and longitudinal acoustic (LA) [transverse acoustic (TA)] phonons with the e1hh1(2s) exciton serving as an intermediate state. The central line is due to elastic scattering of the 2s exciton to the hot 1s-exciton state on a static random potential of the heterostructure followed by emission of a LO phonon.