A mechanism for three-photon double ionization of atoms by extreme-ultraviolet free-electron laser pulses is revealed, where in a sequential process the second ionization step, proceeding via resonant two-photon ionization of ions, is strongly enhanced by the excitation of ionic autoionizing states. In contrast to the conventional model, the mechanism explains the observed relative intensities of photoelectron peaks and their angular dependence in three-photon double ionization of argon.