If you had a car for one day in Siracusa

Noto and Modica.
If you have a car, you can explore a lot more places. The itinerary you have you stay stuck in cities for way longer than is necessary to see those places. 3 nights in Taormina gives you time to explore the area of Mt Etna too.
See how many things you can add.
https://www.visitsicily.info/en/
And make sure you return the car before you spend your time in Palermo. You won’t need a car while in Palermo.

We actually are planning to take an organized tour to Mt Etna which is why we are staying three nights in Taormina.

I actually have come across the web site you kindly provided a link to. It seems to be organized by type of site not area and thus a bit challenging for me to figure out even where some of the places are. Are there other places you would recommend that are reasonable day trips from Siracusa?

We do plan to go to Noto and Modica, but Modica probably from Ragusa.

Villa Romana del Casale.

I liked Noto, but the mosaics at the Roman Villa are the best I have ever seen

Aside from Noto and maybe Avola I don’t remember too many more places around Siracusa that are an absolute must. Modica is certainly closer to Ragusa, so that is best visited from there. On the way to Agrigento you could swing by Piazza Armerina (Villa del Casale) and Caltagirone. It’s a detour but worthwhile. On the western side, after Agrigento, on the way to Palermo you could swing by Selinunte, Segesta, Erice, Trapani, San Vito lo Capo, Scopello, Zingaro Natural Reserve, Castellammare, just to name a few. If it were my trip I would probably borrow one night from Ortigia to spend a night near the places I mentioned. Ragusa probabl can be done in one full day too (2 nights). I would try to find a couple of nights to devote to the western area,

The idea of moving a couple nights to go to Western Sicily is worth considering. I would like to see Erice and the Salt mines in particular. My husband and I did a series of short stays in Crete this past May (9 hotels in 15 nights) mostly successfully. But my daughter and her husband are joining us this time. So a bit more complicated.

Adding Villa Romana is a must imo. Moving a night from Siracusa (or Ragusa really) and staying in the area would ensure you could visit the site at your leisure and avoid crowds. One night stays aren’t ideal, but appropriate here.

Yes, that actually was my original plan but made for two one night stays in a row which my husband did not like. Maybe I will reconsider....

We enjoyed a night in Marsala. It’s small and was great for relaxing. It has no must see sights. En route to Erice we stopped at both the Marsala and Trapani salt farms. Both are off of extremely rural roads that are not well marked. We liked the Marsala farm better than the Trapani one, but neither takes long to see.

We traveled to Sicily in October and followed an itinerary similar to yours (in reverse). I agree that Villa Romana del Casale is worth the time to see. Leave Ragusa and head to Villa Romana del Casale arriving late morning. Spend 1.5 hrs there and then drive to Agrigento. Spend the night in Agrigento and go to Valley of the Temples first thing in the morning. Spend 2.5 hrs there and then drive to Palermo arriving late afternoon. One of our days in Palermo we took the train to Cefalu...it was a fantastic day trip. I can also recommend the Streaty Food Tour in Palermo!

Beth, my favorite spot in Sicily was Erice, but I spent two nights there. The best times were early morning and evening, very misty and mysterious, very few people. Meals were great too, including wonderful Couscous di Pesce, originally a dish from nearby Africa (Tunisia), I believe. Many visitors just go to Erice as a day trip, often much more crowded with tourists, much less misty and mysterious. Also, the nearby Greek ruins at Segesta are beautiful and intact, and sort of out in the middle of nowhere, very picturesque. Segesta is located on your way to Palermo from Erice, although when I was there, I was advised not to park at Segesta with my baggage in the car, as at that time, the parking lot was open space, not well patrolled, and sometimes a target of baggage thieves. Maybe that is different now.