Advice from above is all good and to the point. My wife and I walked from Switzerland to Rome in the fall of 2021, covering the Lucca - Rome section from late October into mid-November. Separately, we also did the Lazio sections into Rome in December, 2018. November, in general, should be a good time to walk, as there is little shade in Tuscany, and October can still be blisteringly hot. Though, of course, the days will be much shorter, with sunset before 5PM. For weather averages, my go-to site is
www.weatherspark.com. Just plug in random cities and the month you're interested in.
The best site, in general, for accommodations is
www.gronze.com, the Bible of the Spanish caminos, which now also covers the Via Francigena from Switzerland to Rome. Open in Chrome browser for English translation. Gronze is very up-to-date with opening and closing dates, and also tells you how far the lodgings are from the Via, which is important in these heavily-touristed regions, where there is plenty of lodgings but often not near the trail. It also has reviews. Way more useful for pinning down accommodations than the guidebooks or the official Via Francigena app. Like Samos, above, we also enjoyed our monastery stays.
If you'd like our itinerary, with suggestions on (possibly) by-passing unattractive stretches, let me know. I also put together a logistical / resource guide which covers Switzerland from Lake Geneva to Rome.