With advancements in robotics, flexible robotic arms have increasingly become part of daily life, assisting or even replacing humans in various tasks across diverse environments due to their inherent safety and flexibility. To enhance the grasping capability of robotic arms and achieve a dynamic and wider grasping range, this study introduces the design, simulation, and testing of a two-step expansion flexible gripper featuring inflatable fingers and a pneumatic palm. The gripper's operation is supported by a pneumatic drive control system that is implemented using a solenoid valve. Compared to existing designs, the proposed gripper and corresponding two-step expansion control enable the gripper to effectively grasp objects with varying geometries and material hardness, offering improved adaptability and functionality. This paper is an extended version of [1].