This paper presents a novel design approach to soft robotic fingers based on the Fin Ray effect, which mimics the flexible yet adaptive behavior of fish fins. These soft, versatile grippers are ideal for applications requiring gentle manipulation and environmental adaptability. The study focuses on integrating multi-stage stiffening mechanisms into 3D-printed thermoplastic polyurethane (TPU) fingers. These mechanisms, featuring sequentially engageable inner structures, allow a multi-stage grasping load via increased deformation. By combining experiments with simulations, this work demonstrates how these inner structures improve the gripper's functionality and extend its application potential in soft robotics.