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The magnetic dipole and the spectroscopic quadrupole moments of the nuclear ground states in the odd-mass nickel isotopes 59−67Ni have been determined using collinear laser spectroscopy at the CERN-ISOLDE facility. They are compared to ab initio valence-space in-medium similarity renormalization group (VS-IMSRG) calculations including contributions of two-body currents as well as to shell-model calculations. The two-body-current contributions significantly improve the agreement with experimental data, reducing the mean-square deviation from the experimental moments by a factor of 3 to 5, depending on the employed interaction. For all interactions, the largest contributions are obtained for the 52− (72−) isotopes 65Ni (55Ni), which is ascribed to the high angular momentum of the f orbitals. Our results demonstrate that the inclusion of two-body-current contributions to the magnetic moment in an isotopic chain of complex nuclei can be handled by the VS-IMSRG and can outperform phenomenological shell-model calculations using effective g-factors in the nickel region.