As can be seen in the famous photos of Jeffrey Hunter and Roddenberry examining the model on-set, the three-footer was built during production of the pilot episode. It was filmed for all of the pilot’s visual effects shots of the ship, EXCEPT for the zoom-in to the Bridge that kicks the story off, which used the 11-footer.
And, of course, several of those three-footer flybys from the pilot were reused in the opening credits of TOS proper—the shots where the ship whizzes by in-between each credit.