The intensity of the Cosmic UV background (UVB), coming from all sources of ionizing photons such as star-forming galaxies and quasars, determines the thermal evolution and ionization state of the intergalactic medium (IGM) and is, therefore, a critical ingredient for models of cosmic structure formation. Most of the previous estimates are based on the comparison between observed and simulated Lyman-α forest. We present the results of an independent method to constrain the product of the UVB photoionization rate and the covering fraction of Lyman limit systems (LLSs) by searching for the fluorescent Lyman-α emission produced by self-shielded clouds. Because the expected surface brightness is well below current sensitivity limits for direct imaging, we developed a new method based on 3D stacking of the IGM around Lyman-α emitting galaxies (LAEs) between 2.9 < z < 6.6 using deep MUSE observations. Combining our results with covering fractions of LLSs obtained from mock cubes extracted from the EAGLE simulation, we obtain new and independent constraints on the UVB at z > 3 that are consistent with previous measurements, with a preference for relatively low UVB intensities at z = 3, and which suggest a non-monotonic decrease of ΓH i with increasing redshift between 3 < z < 5. This could suggest a possible tension between some UVB models and current observations which however require deeper and wider observations in Lyman-α emission and absorption to be confirmed. Assuming instead a value of UVB from current models, our results constrain the covering fraction of LLSs at 3 < z < 4.5 to be less than 25 per cent within 150 kpc from LAEs.