A method for making a lithographic printing plate precursor is disclosed comprising the steps of a. providing a lithographic printing plate precursor including a support having a hydrophilic surface or which is provided with a hydrophilic layer, a photopolymerizable coating comprising a photopolymerizable composition, and an overcoat layer comprising a water soluble low molecular weight acid, wherein said overcoat layer has a surface pH-value ranging between 1 to 6; b. image-wise exposing the precursor; c. optionally heating the exposed precursor; d. processing the obtained precursor by applying a gum solution and optionally brushing the precursor, and/or by mounting the precursor on a plate cylinder of a lithographic printing press and rotating the plate cylinder while feeding dampening liquid and/or ink to the coating. The printing plate obtained after imaging and processing in an off-press configuration with a gum solution or in an on-press configuration with fountain solution and ink shows an improved roll-up performance in the start-up of a printing process on a printing press.