If you haven't included Cape Sounion, you might consider taking an EARLY bus there, and getting there by 9 am, the early morning light would be fab for the temple columns. There's a bus that goes straight there. THen you can go to the beach just below & have a swim & picnic before heading back. Most people take tour busses there for the sunset, and then are dismayed to see a dozen busses pulled up, like a herd of elephants at the waterhole, and hordes of people scambling out. YOu wouldn't be able to get a decent photo without crowds in it. But in the AM, if you're a "lark" and up early, it could be fun... and swim is good for your last day too -- that is, if you're going in warm-enough weather. ( Why why why do people not give the dates of their trips if they want the best advice???, mutters the cranky old hand).
Before suggesting something outside of Athens just because it's doable in 6 hours ... why not tell us what "major sites" you will have seen. Some "Minor" places IN Athens may be very meaningful if they speak to personal interests of yours. Just a few examples:
• JEWELRY?? - Ilias LaLounis was THE world-famous jewelry designer of Greece (Jackie O made him famous with Americans); think a combo of Cartier & Tiffany for prestige. His mansion , just 100 meters or so away from Acropolis Museum, showcases designs based on 5 ancient civilizations, and also has demonstrations of goldsmithing.
• JEWRY? - On a quiet sidestreet in Plaka, a quiet townhouse is Athens Jewish Museum... a poignant story of Greece's long history of flourishing Judaism and its brutal ending under the Nazi occupation -- Greek people & towns tried to protect their fellow citizens, in vain. It will only take 30-40 minutes to go through it.... but what you see will linger much much longer.
• ATHENS' BELOVED MELINA -- Melina Mercouri Foundation (over in Monastiraki area) has enthralling photorama of that fabulous Actress' gifts & activism -- how she walked off the stage on Broadway when a coup engulfed Greece and toured the world rallying allies to rout the dictator, & then as Minister of Culture helped educate a nation about its ancient Glories. I found it thrilling!
• CHILDREN'S ART -- Right in PLaka is the Museum of Children's Art -- amazing postcards & prints from annual national contest -- the 2008 winning poster ("My Hometown Athens" by an 8 year old) is framed on my diningroom wall.
• MUSIC?? -- The Popular Instruments Museum, on a high-up street in Plaka, not only displays ancient & modern instruments, you can push buttons & hear just how they sound PS: they have good CDs -- I play my mandolin CD of village songs every Spring when I do "greek night" on my little Philly street
• MYSTERIOUS MARBLES - the Cycladic Museum has those amazing pre-Minoan statuettes you won't see on your cruise ... and it also has a superb 20-minute video on how Ancient Greek children learned their culturel I recommend it to ALL.
... Any one of these can top off the final day of your visit.