Recent work on the Taylor-Couette problem in the infinite-cylinder approximation has revealed that, if the driving velocity of the inner cylinder is not steady but modulates in time, two classes of time-dependent flow exist: reversing flows and nonreversing flows. The latter are at first surprising, since the direction of rotation of the Taylor vortices is decoupled from the (driving) azimuthal flow. Since experiments are performed in cylinders of finite height, the natural question which we address in this paper is whether the Ekman circulation induced by the end walls suppresses the reversing-nonreversing effect. We find that the answer is negative--nonreversing flows are actually favoured--and we reveal a variety of new flow patterns including "side-by-side" vortices.