We have investigated both experimentally and numerically the magnonic band structure of arrays of closely spaced Fe/permalloy nanowires (NWs) with an L-shape cross-section using the Brillouin light scattering technique and GPU-based micromagnetic simulations. NWs consist of a 340 nm wide and 10 nm thick permalloy layer covered by a 170 nm wide Fe overlayer. The thickness of the latter was varied in the range from 0 to 10 nm in order to analyze its influence on the magnonic band structure. We found that both the frequency and the spatial profile of the most intense and dispersive mode, can be efficiently tuned by the presence of the thin Fe NW overlayer. In particular, by increasing the Fe thickness, one observes a substantial frequency increase, while the spatial profile of the mode narrows and moves to the permalloy NW portion not covered by Fe. In addition, the presence of the Fe overlayer causes a significant increase of the number of detected modes and a change of their intensity in the Brillouin spectra as a function of the Bloch wave number. These results show that it is possible to engineer the band structure of magnonic crystals consisting of bi-layered, L-shaped, NWs by a careful control of the overlayer thickness.