Naxos would be a good choice, and it's on the main ferry line from Santorini to Athens. It doesn't have a deep enough port for cruise ships, so it has a whole different vibe. Don't get me wrong, it has plenty of tourists, but not the hordes found on Santorini and Mykonos. It has one large ruin visible and a short walk from the port, and pretty white kastro, and an interior carpeted with olive groves. It's quite a bit different from Santorini and Mykonos, much greener and much larger. There are some archaeological sites, some nice beaches, and some little villages in the interior that aren't a tourist draw and thus are more typical. You may want to rent a car for a day or two to explore.
Paros and Antiparos are very close to Naxos and worth checking out. I haven't been to them, so can't comment.
My favorite Greek island is Rhodes. It's reachable by night ferry from Santorini if your timing is right - it only goes a couple days a week. It was a very comfotable ride and an easy way to get between islands. Rhodes is not in the white and blue village family of islands. It was an important launching site for the crusades and consequently has a fortress (and ruins of other fortresses) in addition to some of the more ancient ruins. It has some beaches I assume are nice, as they were gorgeous from a distance. There's a whitewashed village, Lindos, that's an easy bus ride from Rhodes town. I rented a car for a day and explored the eastern coast - it would have taken at least another day to see the rest of the island. I was there four days, and it definitely wasn't enough.