The diffusion of spin-polarized electrons through a graded-bandgap Cd_1−xMg_xTe layer has been studied. The quaternary compound Cd_0.73Mg_0.24Mn_0.03Te diluted magnetic semiconductor (DMS), with a large energy gap (E_g = ~2 eV), was used as a spin-polarized source in the presence of an external magnetic field. We exploit the spin-dependent feature of trion formation (X⁻_s), i.e. negatively charged excitons in a spin-singlet ground state, as a sensitive probe of electron spin injection into a CdTe quantum well (QW), separated from the injector by a thick non-magnetic graded gap Cd_1−xMg_xTe layer (which bends the band edges of the structure similar to electrical biasing). Continuous wave and time-resolved magneto-photoluminescence and reflectivity measurements have been performed. Our results show that at small magnetic fields the ratio of the density of X⁻_s and that of neutral excitons, X⁰, is large when the light is absorbed only in the QW; it becomes small when the absorption occurs in the non-magnetic graded barrier. It is re-enhanced significantly again when the absorption occurs also in the magnetic layer. The latter finding indicates that electrons do not lose completely their initial spin polarization acquired in the DMS layer (due to the giant Zeeman splitting effect in this layer) when they pass through the non-magnetic region into the QW where they enhance the X⁻_s populations. This re-enhancement occurs in spite of a large energy relaxation (~300 meV) that the carriers must undergo.