You should expect some degree of stiffness for months. My stiffness "moved around" to different parts of my leg and back. One area would improve only to have discomfort pop up somewhere else.So here's my question-- the stiff gait when unaided. How long on average did it take you to lose the stiffness in your upper thigh/groin and lower back? Getting a more normal gait unaided?
That said, every hip is unique -- so much depends upon the condition of those muscles prior to surgery, how much the surgeon had to disturb all your soft tissues during the procedure, and how your body handles the recovery.
Maintaining proper gait as you walk will help. It's better to walk correctly for a short distance a couple of times/day than to walk poorly for long distances. Keep using that cane. Practice proper heel-toe walking. At some point, things will "click" and your gait will be better without the cane than with it.
Don't try to rush this recovery; it will only slow you down.