In this paper, we present a singularity-based resolution of the Gibbs phenomenon that obstructs the reconstruction of a function with jump discontinuities by a truncated Chebyshev series or a Padé-Chebyshev approximation. We tackle the more difficult case where the jump locations are not known. The identification of unknown singularities is carried out using a Padi-Chebyshev approximation. Numerical examples to illustrate the method are provided, including an application on postprocessing computational data corrupted by the Gibbs phenomenon.