It's important to note that it is not possible to FLY from one ISLAND to another (except Crete - Rhodes maybe 1 other), but I do agree with Larry that if your stay is short as 10 days, then maybe an early AM flight back to mainland would be wise. Another tip is to avoid TWO stays in Athens -- the business of taxi/bus/metro from airport into central Athens (60-90 mins), cheching into hotel (30 mins), and then doing the same thing to leave -- can take precious Hours you cannot spare ... so why do it 2x?? Why not save Athens to the last, and instead leave directly from Airport to FIRST island??? THat's the strategy all of us "Greece Vets" advise.
You could save even more time, if you already have booked your interntional flght & know your arrival time in ATH. To get to Mykonos you could ferry from Piraeus or (maybe better) from the seaport of Rafina, a 30 minute taxi from airport. In june there's usually a 3 pm ferry that could get you there at 6 pm or so. ORRRR... since u are planning so far ahead, you might well be able to take a FLIGHT (35-40 mins) vs a Ferry of 3-4 hours -- at a similar cost! Aegean air is beginning to post its "promotional rates" for 1st half of 2018... If you want to get alerts to all the latest discount fares available you have to subscribe to its "newsletter" (which is really an e-mail alert) -- you can do so here: https://aegeanairlines.custhelp.com/app/answers/detail/a_id/302/related/1 I just checked June 9 and there's a 1:25pm flight (35 mins) for €67.94. (including 1 big checked bag).
If you fly, you still will be able to have the "Greek ferry experience" between Mykonos & Naxos. Getting back from Naxos, flights are less frequent ... mainly early AM or late morning -- and on smaller planes that fill up FAST when bargain fares are announced. If you choose to go by ferry, the BLUE STAR line is the best choice, because they have modern LARGE ferries with outside decks, and comfy inside "reserved economy" seats for the 4.5- 5 hr journey. The daytime run (9:45 - 3) eats up a whole day ... a way to give yourself a last wonderful afternoon on the beach is to take what I call the "sunset sailing" (6pm - 11:30pm). You'd have to check out of your hotel @ noon of course, but hoteliers will willingly stash your bags in a storage room, and then let you use a lobby bathroom late in the day for a quickie wash & change into travel clothes.
Either way, you get to Athens LAST, after you're already more accustomed to Greek ways, and revitalized from your island time -- energized to make the most of sightseeing in a vibrant and historic major city. You're already on the road to a wonderful trip, by thinking ahead.
PS: in June really no need to book ferry tickets ahead, unless for Friday or Mon night when its a 3-day holiday weekend ,,, and there is none in 2018. You just buy the day of or day before, depending, at any ticket agency.