Collisionless shock waves are efficient ion accelerators. Previous numerical and observational studies have shown that quasi-parallel (Q∥) shocks are more effective than quasi-perpendicular (Q⊥) shocks at generating energetic ions under steady upstream conditions. Here, we use a local, 2D, hybrid particle-in-cell model to investigate how ion acceleration at super-critical Q⊥ shocks is modulated when tangential discontinuities (TDs) with large magnetic shear are present in the upstream plasma. We show that such TDs can significantly increase the ion acceleration efficiency of Q⊥ shocks, up to a level comparable to Q∥ shocks. Using data from the hybrid model and test particle simulations, we show that the enhanced energization is related to the magnetic field change associated with the discontinuity. When shock-reflected ions cross the TD during their upstream gyromotion, the sharp field change causes the ions to propagate further upstream, and gain additional energy from the convection electric field associated with the upstream plasma flow. Our findings illustrate that the presence of upstream discontinuities can lead to bursts of energetic ions, even when they do not trigger the formation of foreshock transients. These results emphasize the importance of time-variable upstream conditions when considering ion energization at shocks.