We consider hydromagnetic Couette flows in planar and spherical geometries with strong magnetic field (large Hartmann number, ). The highly conducting bottom boundary is in steady motion that drives the flow. The top boundary is stationary and is either a highly conducting thin shell or a weakly conducting thick mantle. The magnetic field, , is a combination of the strong, force-free background and a perturbation induced by the flow. This perturbation generates strong streamwise electromagnetic stress inside the fluid which, in some regions, forms a jet moving faster than the driving boundary. The super-velocity, in the spherical geometry called super-rotation, is particularly prominent in the region where the ‘grazing’ line of has a point of tangent contact with the top boundary and where the Hartmann layer is singular. This is a consequence of topological discontinuity across that special field line. We explain why the magnitude of super-rotation already present when the top wall is insulating [Dormy, E., Jault, D., Soward, A.M., 2002. A super-rotating shear layer in magnetohydrodynamic spherical Couette flow. J. Fluid Mech. 452, 263–291], considerably increases when that wall is even slightly conducting. The asymptotic theory is valid when either the thickness of the top wall is small, and its conductivity is high, ɛ or when and ɛ. The theory predicts the super-velocity enhancement of the order of in the first case and ɛ in the second case. We also numerically solve the planar problem outside the asymptotic regime, for ɛ and , and find that with the particular that we chose the peak super-velocity scales like . This scaling is different from found in spherical geometry [Hollerbach, R., Skinner, S., 2001. Instabilities of magnetically induced shear layers and jets.