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Maser lines from different molecular species, including water, hydroxyl, and methanol, are common observational phenomena associated with massive star forming regions. In particular, the methanol maser appears as an ideal tool to study the early phases of massive star formation. However, it is difficult to establish the exact start of the methanol maser phase, and it would then be interesting to detect and study low-flux density methanol masers (i.e., < 0.1 Jy or even << 0.1 Jy), in order to determine if they can effectively be used to mark a specific evolutionary phase in high-mass star formation. Past surveys have been unable to systematically detect many low-flux density methanol masers, and thus we do not yet know how many such masers exist in the Galaxy and what is their physical nature. Out of a sample of 107 observed Hi-GAL sources we detected a total of 32 methanol masers, with 22 sources being new and weak (median peak flux density 0.07 Jy) detections, in the Galactic longitude range [32.0, 59.8]deg. We also detected 12 6.035-GHz OH maser, with 9 objects being new detections. Our survey covers a similar range of source distances as the "Arecibo Methanol Maser Galactic Plane Survey", but the methanol masers detected by us are clearly shifted towards lower integrated flux densities. The newly detected methanol masers are mostly of low-luminosity and, except for some sources, their weakness is not due to distance effects or positional offsets. No specific correlation is found with the physical parameters of the Hi-GAL clumps, except for sources with both CH3OH and OH masers which tend to have higher mass and luminosity. The intensity of the methanol masers correlates well with the velocity range of the maser emission, which suggests that the low brightness of these masers is related to the number of maser spots in the emitting region and their evolution with time., Comment: This paper contains a total of 14 figures and 7 tables. Submitted for publication to A&A on November 4th, 2013