We present a theoretical framework and validation for manipulating instability growth in shock-accelerated dual-layer material systems, which feature a light-heavy interface followed by two sequential heavy-light interfaces. An analytical model is first developed to predict perturbation evolution at the two heavy-light interfaces, explicitly incorporating the effects of reverberating waves within the dual-layer structure. The model identifies five distinct control regimes for instability modulation. Shock-tube experiments and numerical simulations are designed to validate these regimes, successfully realising all five predicted states. Notably, the selective growth stagnation of a perturbation at either the upstream or downstream heavy-light interface is realised numerically by tuning the initial separation distances of the three interfaces. This work elucidates the critical role of the wave dynamics in governing interface evolution of a shocked dual layer, offering insights for mitigating hydrodynamic instabilities in practical scenarios such as inertial confinement fusion.