Cervical spinal cord injury (SCI) can severely impact hand motor and sensory function, and accordingly, patients with SCI are often unable to complete basic everyday tasks without assistance. In recent years, there has been an increase in hand exoskeleton research due to their distinct advantages for improving rehabilitation. In this work, we present a voice-controlled, tendon-driven soft robotic hand rehabilitation exoskeleton for hand function improvement in patients with cervical SCI. A new fabrication process utilizing high consistency rubber silicone is used to construct a formfitting, durable, and customizable exoskeleton glove. Bioinspired tendon routing pathways embedded within the glove create 5 actively actuated degrees-of-freedom in the index finger, middle finger, and thumb for both extension and flexion. Tendon tension sensors are developed and integrated into the exoskeleton system. The force feedback from the sensors is used in the implementation of admittance control for the exoskeleton system to improve grasping motions. The exoskeleton was evaluated in a case study with a healthy participant through range-of-motion characterization, pinch force testing, admittance controller validation, and object manipulation.