Late September is a great time to visit Greece. It is still warm the sea is warm but you might run into a day or so of rain.
Book that $560 fare.
Experienced travelers to Greece generally will suggest you get out to your first island immediately on landing in Greece. You will need a two hour window for the connecting fight. It is way better getting over jet lag on an island rather than Athens. Also it is a really good idea to be in Athens at the end of your trip so you are on the mainland for your return flight. Stuff can happen especially when you are trying to get back from an island for a same day flight. We were on a fight that got delayed 4 hours. 5 couples missed their same day return flight. Why take that chance.
So I would fly to the furthest island which would be Santorini. There are two or three evening flights for you to catch. Personally I think 3 nights is probably a minimum on Santorini. 2 nights is really only a day and a half.
Now ferry to Naxos. Naxos is a great companion island with Santorini. It will provide you with an authentic Greek experience. It has great beaches, lovely main town topped by a Venetian era Kastro, mountain villages and all of this at about half the price of Santorini. Naxos needs 5 to 7 days to visit properly.
I am now going to diverge from your proposed plan. By visiting two Cyclades island Santorini and Naxos you will have had a great island trip. There is absolutely no reason to include Hydra. It is nice and is recommended by Rick but that is because his Greek tours do not include an island so Hydra is intended to give you a little taste of an island. No need to do that after Naxos. I am also going to suggests your skip Mykonos. It too is beautiful but it markets itself as a high end jet setter destination. That means Santorini prices for meals and hotels and shopping but all this without the Caldera views. Naxos has everything Mykonos has but at half the price.
Another reason to cut your destinations down is because every time you change locations you lose a minimum of half a day. So each of your 2 day locations is really a day and a half. By dropping Hydra and Mykonos you now have 5 nights in Naxos and 3 in Santorini.
Fly back to Athens. Pick up your rental car. We use ACR Athens car rental. They are highly recommended. Drive to Nafplio. On the way stop in Corinth and visit the Canal, Ancient Corinth and Acrocorinth then Ancient Nemia. It has sister games to Olympia and is a tidy easily accessible site. Now go to Nafplio. It is a lovely seaside town. I can recommend hotel Agamemnon.
Nafplio needs a couple of days and a couple more to visit the ancient sites nearby. There is almost no way you can do Olympia and Monemvassia in a 4 day Nafplio segment. Olympia is a 5 hour drive each way. Monemvassia is probably 2.5 to 3.5 hours each way. Substitute Nemia for Olympia. You indicated you could add some time to your trip. I would suggest you consider 5 to 7 days for the Peoponnese including a night in Monemvassia. Its amazing to sleep in a Medieval walled town. Don't miss Mystras on the way to Monemvassia. Its the last Byzantine city.
Finish with your time in Athens.
Here is what I am proposing
Naxos Town http://www.flickr.com/photos/stanbr54/sets/72157632094558042/
Trip around Naxos
Naxos trip 2019 https://www.flickr.com/photos/stanbr54/albums/72157711294807761
Corinth area. https://www.flickr.com/photos/stanbr54/albums/72157711755443483