Bistable structures have been widely utilized in soft grippers to reduce the energy required for maintaining grip. Grippers have been investigated in terms of the energy efficiency and accuracy of gripping; however, the limited number of gripping states hinders the holding of objects of various shapes. In this study, an energy-efficient gripper was developed to accommodate both convex and concave shapes using a tristable structure that combines two bistable structures, with shape memory alloy wires used as actuators. Different gripping modes were designed for convex and concave shapes, based on three states of the gripper: gripping, open, and holding. The gripper consisted of a driving part with a leaf spring for a “linear snap action” and a soft finger part with an elastic ring and prestressed fingers. Geometric variables were adjusted to construct a tristable energy curve through experiments and analyses. The fabricated gripper weighed about 140 g and was capable of gripping convex objects of up to 80 g, and concave objects of about 120 g. Only a small amount of energy was consumed in the switching states, and the gripper maintained a stable state while gripping with no energy consumption. It is expected that this research will contribute to lightweight and energy-efficient grippers for application to drones, for example.