At transonic flight conditions, shock oscillations on wing surfaces are known to occur and result in degraded aerodynamic performance and handling qualities. This is a purely flow-driven phenomenon, known as transonic buffet, that causes limit cycle oscillations and may present itself within the operational flight envelope. Hence, there is significant research interest in the development of shock control techniques to either stabilise the unsteady flow or raise the boundary onset. This paper explores the efficacy of dynamically activated contour-based shock control bumps within the buffet envelope of the OAT15A aerofoil on transonic flow control numerically through unsteady Reynolds-averaged Navier–Stokes modelling. A parametric evaluation of the geometric variables that define the Hicks–Henne-derived shock control bump will show that bumps of this type lead to a large design space of applicable shapes for buffet suppression. Assessment of the flow field, local to the deployed shock control bump geometries, reveals that control is achieved through a weakening of the rear shock leg, combined with the formation of dual re-circulatory cells within the separated shear-layer. Within this design space, favourable aerodynamic performance can also be achieved. The off-design performance of two optimal shock control bump configurations is explored over the buffet region for $M=0.73$, where the designs demonstrate the ability to suppress shock oscillations deep into the buffet envelope.