This paper presents the SoftHand Model-W: a 3D-printed, underactuated, anthropomorphic robot hand based on the Pisa/IIT SoftHand, with an integrated antagonistic tendon mechanism and 2 degree-of-freedom tendon-driven wrist. These four degrees-of-acuation provide active flexion and extension to the five fingers, and active flexion/extension and radial/ulnar deviation of the palm through the wrist, while preserving the synergistic and self-adaptive features of such SoftHands. A carpal tunnel-inspired tendon routing allows remote motor placement in the forearm, reducing distal inertia and maintaining a compact form factor. The SoftHand-W is mounted on a 6-axis robot arm and tested with two reorientation tasks requiring coordination between the hand and arm's pose: cube stacking and in-plane disc rotation. Results comparing task time, arm joint travel, and configuration changes with and without wrist actuation show that adding the wrist reduces compensatory and reconfiguration movements of the arm for a quicker task-completion time. Moreover, the wrist enables pick-and-place operations that would be impossible otherwise. Overall, the SoftHand Model-W demonstrates how proximal degrees of freedom are key to achieving versatile, human-like manipulation in real world robotic applications, with a compact design enabling deployment in research and assistive settings.