The frustrated magnet SrDy2O₄ exhibits a field-induced phase with a magnetization plateau at 1/3 of the saturation value for magnetic fields applied along the b axis. We report here a neutron scattering study of the nature and symmetry of the magnetic order in this field-induced phase. Below T ≈ 0.5 K, there are strong hysteretic effects, and the order is short- or long-ranged for zero-field and field cooling, respectively. We find that the long-range ordered magnetic structure within the zigzag chains is identical to that expected for the one-dimensional axial next-nearest neighbor Ising (ANNNI) model in longitudinal fields. The long-range ordered structure in field contrasts with the short-range order found at zero field, and is probably reached through enhanced quantum fluctuations with increasing fields.