You start in Palermo, then finish in Catania (April 20), with the Off-Season version of the Best of Sicily tour, correct? Then you meet up in Rome for the next tour, which ends in Naples.
The first tour does NOT include Taormina at all... not... nope!
Yet, by mid-April, Taormina is already in full swing for receiving visitors. IMO, Taormina is NOT to be missed for a Spring/Summer/Fall visit to Sicily. The nice thing about the end of April is that it is not too busy yet.
It would be a real shame to go to Sicily and miss staying in Taormina for a couple of days. Catania is a good place for doing a train or bus daytrip from (to another of the 'Hilltop' towns that your guide skipped on Day 8 - like Modica, Noto, Sciccli, etc. - or to Acireale or even back to Siracusa for the day). Also, bad weather can impact your Tour day to Mt. Etna - you can then have a back-up plan on going back from either Catania, or just as easily from Taormina.
Assuming that you have 5 nights, here is what I would do in your shoes:
1 night - book one extra night in the Tour's hotel in Catania.
- plan a day trip on your own, or book a tour.
- Post a question on the Italy Forum for "Catania Daytrip Suggestions"
3 nights - Taormina.
- Take the direct bus from Catania to Taormina.
- Post a question on the Italy Forum for "Taormina Suggestions" (hotels, etc.)
1 night - journey to Rome (see options 1 & 2)
1 night - book one extra night in the next tour's hotel in Rome
- Take the direct bus from Taormina to Catania Airport
- Take a midday/early afternoon flight from Catania to Rome
1 night - book the overnight direct train from Taormina to Rome Termini in an 'Excelsior Doppio' private sleeper cabin for two. For April, you would be able to book this in February.
I like your idea of spending extra days in Sorrento. However, given what is covered in your tour, you really wouldn't need many extra days... maybe 2?? It would make much more sense to add these in at the end, since you do finish in Naples (or stay 2 extra nights in Naples itself). To me, this makes more sense, instead of detouring to Sorrento in the middle IMO.
Have fun figuring it all. Seriously though, your focus should be Sicily for the extra nights.