Vienna

It looks like the tour does not go to Melk Abbey so I would highly recommend that as a day trip from Vienna - train to Melk, Danube cruise to either Krems or Durnstein, then return to Vienna. It was the best day trip I took in that area.

You will have to decide how much energy you have for more places versus more sites within Vienna.
Melk is very good for extra travel. There is a Danube-side bicycle path and you can rent electric bikes easily and just head off and return for something more athletic. Heurige wine gardens south of town are easily reachable by bus or tram and are a fun afternoon/night out (get recommendations from your guide). You will go to the Hapsburg Treasury, but the Kunsthistorische museum showcases how much swag rich European emperors could have their minions haul back home from places like Egypt, etc.throughout the 19th Century. (think rooms and rooms of mummies, etc.) I like Schoenbrunn Palace better than the Hapsburg Palace in town, but there are lots of sites to pick from. The Belvedere Palace south of town has Gustav Klimt's "The Kiss" and other excellent more modern art. Since you are staying a few extra days you can try to book other music events that the tour couldn't arrange.

And then there is more chocolate cake.

One of my favorite places in Vienna is the Dorotheum. It's the largest antiques auction house in Europe, and was started by the Holy Roman Emperor as a pawn shop for his nobles to get cash. One of my favorite souvenirs is a gold pocket watch from a race in 1904.

I’m confused. Are you planning to stay in Vienna 3-4 more days or are you looking for another location to visit in the region for 3-4 days?

Take the 2.5 hour train to Budapest.

Oh James, that was my first inclination too. But I can't imagine just a day trip to Budapest, it would be like taking one bite of a marvelous dinner and then walking away from the table. :)