Palermo is chock full of very interesting churches and oratorios. Hours vary, so you can usually find something open on a day when you have free time. With a good guide book and a city map you can identify some geographically-convenient possibilities for the morning of Day 1 and the afternoon of Day 2. I also really enjoyed the tile museum. I've linked to TripAdvisor because the museum's web site is in Italian only. There was an English-speaking guide when I visited in June 2015. You must phone ahead to make a reservation.
If you'd prefer a picturesque costal town (quite touristy), there's rail service to Cefalu, which is east of Palermo. If you arrive a day early, Cefalu would be easy to visit on Day 1. Be sure, though, that it's not part of the tour; I remember that Rick had very positive things to say about it on the Sicily video.
On Days 3 and 4 I'd spend any free time just strolling around Trapani's historic district, since the tour takes you to Erice.
On Day 7 in Siracusa, I recommend the Archaeological Park and Greek Theatre--especially nice because the former has lots of greenery--and the archaeological museum. Ancient ruins are definitely not my thing, but I really liked the park. All of the above are far enough from Ortygia that you'd probably prefer to take a bus or taxi. If you've already overdosed on Greek ruins at Agrigento, you can instead spend additional time exploring Ortygia; it's rather large, and there are many picturesque streets to wander.
In my view, the main thing missing from the tour that can't be squeezed into any of your free half-days is a trip to one or more of the baroque towns in the southeast, including Ragusa, Noto, Modica and Scicli. With an extra day or two on Sicily it would be possible to reach some of them by public transportation. There are also some tours offered--not necessarily every day of the week--from Siracusa and Taormina; probably also from Catania, but I haven't checked that.
Edited to add: Sicily has fabulous food in general (quite different from standard Italian), but is especially notable for the world's best granite (dairy-free frozen ices). Don't be suckered in by a spot that just squirts syrup over shaved ice. You want to buy at a place that has a separate dispenser for every flavor it sells (which might be only one or two). The almond is unusual and very refreshing. A special combination sometimes found in Catania but perhaps native to Noto is almond with a small dab of coffee ice. I don't remember what that combo is called. Perhaps "xxxx catanese"?
A breakfast option you may encounter gelaterie and cafes is the brioche + granite combination. The two components were always served to me separately, but I've heard of ice cream sandwich-like offerings.