The interaction between non-premixed lifted flames and the free jet structures is of paramount importance in the stabilization of combustion processes. So, by organizing or disorganizing the jet structures, acoustics must also act on flames by its ability either to amplify the flames' intrinsic drawbacks, or to change and improve the combustion regime. An earlier systematic study, carried out by the authors, concerns the responses of a non-premixed flame, lifted in its hysteresis zone, to sine-shaped forcing in a wide range of frequencies and amplitudes, under fixed conditions of burner diameter and flow rate. The present work is aimed at verifying the robustness of the physical mechanisms involved over a wider range of conditions. These conditions are chosen in a range of moderate Reynolds numbers (3000-6000) so that the flame stabilizes in the hysteresis zone. The influence of imposed oscillations on the flame is summarized in charts as a function of the forcing frequency and the fluctuation amplitude. Results show that the phenomena presented previously are still observed, but shifts of the response domain limits are noticed. Some objective global criteria are proposed to explain these shifts. A particular behavior (the "split-flame'' described by Hertzberg et al.) is only observed for the case of small diameters.