I like to explore, so my recommendation will be different. You will be going to Orvieto, the Tuscany region & Venice on your tour. I would suggest taking the train your next morning to Salerno. Take the Frecciarosso (fast train) options on Trenitalia - an easy, no train connections route for you, and you will be at Salerno by 10:30am if you take the 8:58am train from Roma Termini. If you decide to do this, purchase your train tickets a month ahead to save money.
From Salerno you can take ferries or the bus along the Amalfi Coast for a day or two of sightseeing. Positano & Ravello are my two favorites. Or, go the opposite direction one day and see the amazing Paestum Greek ruins. (Google photos of them!) It’s a 30-minute train from Salerno, and the ruins are a very short walk from the train station. This train is a cheap one - no need to buy a month or even a week ahead.
Those options would be a nice complement to the rest of your trip since the rest of your trip is inland except for Venice.
Full disclosure- I’ve stayed in 40 cities in Italy (yes, I love Italy!), and I think all of those pre-tour days spent in Rome would be an expensive choice & more intense than I would suggest if a friend was asking me for advice.