We demonstrated experimentally a direct way to probe a hidden propensity to the formation of a spin-density wave in a nonmagnetic metal with strong Fermi surface nesting. Substituting Fe for a tiny amount of Cu (1%) induced an incommensurate magnetic order below 20 K in heavily overdoped La(2-x)Sr(x)CuO(4). Elastic neutron scattering suggested that this order cannot be ascribed to the localized spins on Cu or doped Fe. Angle-resolved photoemission revealed a strong Fermi surface nesting inherent in the pristine La(2-x)Sr(x)CuO(4) that likely drives this order. Our finding presents the first example of the long-sought "itinerant-spin extreme" of cuprates, where the spins of itinerant doped holes define the magnetic ordering ground state; it complements the current picture of cuprate spin physics that highlights the predominant role of localized spins at lower dopings.