The smallest member of the Didymos binary near-Earth object system (Dimorphos) is the target of the DART/LICIACube mission, the first attempt to change the orbit of another celestial body via a kinetic impactor. It is important to characterize the unperturbed system prior to the DART impact. In this work we obtained, for the first time, spectral characterization of the system at several rotational phases from TNG+DOLORES in the visible range (0.34-0.81 μm). This is crucial in order to disentangle the primary and secondary bodies and highlight eventual dishomogeneities on their surfaces. We confirm that a subtle but persistent spectral variability appears, even when compared with data obtained from previous 2003 and 2019 apparitions. While the reason for such variability is still under investigation, our analysis hints that different compositions could play a role. Future observations during the brighter 2022 apparition in synergy with data obtained from LUKE on board LICIACube will definitely tackle this conundrum. *Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the Istituto Nazionale di Astrofisica (INAF) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias (Program AOT42-TAC8).