In daily living, people grasp an object through the steps of "pre-shaping" and "enclosing," with the thumb playing a crucial role with its multiple degrees of freedom. When assisting individuals with hand impairments using soft wearable robots, it is important to simplify the robot by reducing the number of actuators and to provide different grasping strategies based on various objects being handled. In this article, we propose a tendon-driven soft wearable hand robot, Exo-Glove Poly III, that uses a single actuator for assisting two types of grasping strategies for people with hand impairment. To move the thumb and other fingers with a single actuator, we developed a slack-based sequential mechanism that allows movements to occur at different timings by varying the initial slack lengths of each tendon. Based on our observations of grasping strategies and the proposed novel actuation system, a slack-based sequential actuator (318 g, including electronic circuits) was designed and integrated with the glove (90 g) using a commercial armband to make the system portable. The robotic system was evaluated by a healthy subject, showing how the thumb moves by the tendon routings and how the mechanism works for each grasping strategy.