Exceptional points (EPs) have extensive and important applications in many wave-based technologies, such as ultra-sensitive sensing, unidirectional scattering and low-threshold laser. However, most of the previous EP-related wave phenomena are demonstrated in systems with fixed configuration, thereby extremely constraining their adaptability and reconfigurability in practice. Here, we introduce a flexible approach to tuning EPs in an acoustic system with sandwich structures. A rotatable component, associated with an alterable gradient index, is clamped by a pair of lossy acoustic resonators. Theoretical derivations and numerical simulations validate the capabilities of the model in continuously regulating EPs in the parameter space, with ingenious experimental setups confirming these findings. The results showcase the system’s effectiveness in achieving unidirectional reflectionless wave propagation across various frequencies. Our research reveals a flexible approach to linking the adjustment of EPs to a simple structural parameter, offering a robust framework for exploring and implementing non-Hermitian wave phenomena in practical scenarios.