Final answer:
Osteoarthritis of the knees is the most likely cause of the client's increased pain following a 3-mile walk.
Step-by-step explanation:
The most likely cause of the client's increased pain following a 3-mile walk is Osteoarthritis of the knees. Osteoarthritis is a degenerative joint disease that causes pain, stiffness, and swelling in the joints, particularly in the knees. The prolonged walking may have put strain on the affected knee joints, leading to increased pain.