We explore magnetic order in the quantum spin chain compound SrCo₂V₂O₈ up to 14.9 T and down to 50 mK, using single-crystal neutron diffraction. Upon cooling in zero-field, commensurate antiferromagnetic (C-AFM) order with modulation vector k_C = (0, 0, 1) develops below T_N ≃ 5.0 K. Applying an external magnetic field (H∥c axis) destabilizes this C-AFM order, leading to an order-disorder transition between T_N and ~1.5 K. Below 1.5 K, a commensurate to incommensurate (IC-AFM) transition occurs at 3.9 T, above which the magnetic reflections can be indexed by k_IC = (0, 0, 1 ± δl). The incommensurability δl scales monotonically with H until the IC-AFM order disappears around 7.0 T. Magnetic reflections modulated by k_C emerge again at higher fields. While the characters of the C-AFM, IC-AFM and the emergent AFM order in SrCo₂V₂O₈ appear to fit the descriptions of the Néel, longitudinal spin density wave and transverse AFM order observed in the related compound BaCo₂V₂O₈, our results also reveal several unique signatures that are not present in the latter, highlighting the inadequacy of mean-field theory in addressing the complex magnetic order in systems of this class.