As I mentioned in reply to your other post, I would do a food tour, either with SRM if it is offered in conjunction with the RS tour or via another tour company (plenty of suggestions in the forum). I'd also go to the Blue Mosque early in the morning just after the second prayer for a quieter experience. If you're particularly interested in something on the printed itinerary and don't mind paying out of pocket, then make time to go see it. Other things not included you might want to see on your own time: Basilica or Theodosius cisterns, Archaeological Museum, Modern Art Museum, mosaics museum (was closed wehn I was there), Galata tower, Rustem Pasha mosque (beautiful), a proper Turkish bath/hammam. It looks like the tour no longer goes to Süleyman mosque - it's a bit off the path but I enjoyed seeing it. I'm not a shopper, so the time in the grand bazaar and the spice market was too much for me, but if you like shopping you might want to go there on your own time. The only place I felt like maybe I was rushed a bit was Topkapi Palace, but I was also super tired and probably wouldn't have spent more time than we did as a group.
FWIW, our guide didn't follow the printed itinerary for Istanbul in the order listed - we did everything just not on the days listed. You will get a Turkiye museum pass when your tour starts, so if you don't want to pay out of pocket for something covered by it, wait to fit that in to free time on your second day if you can (you won't have much free time, though).