Hey, welcome. We were at Misawa for three years, and loved it. Be sure to check out northern Japan while you're there - it is totally different from the Tokyo area.
I would check out what they can offer you at ITT. Sometimes they get some awesome deals, though August might be a bit dicey. There were several travel agencies off base at Misawa that catered to the American market there - most of the flights were the multiple stop ones to the US, but they had flights elsewhere, as well. Someone at ITT might be able to direct you to places around you that might have good deals.
Hops won't work for you, from the sound of it, bit too touch and go for such a long trip.
If you decide that Europe is out of the question (maybe you can get stationed there and do you're traveling then), there is so much in Asia, and even just in Japan to explore. A month would give you plenty of time to investigate the different regions of Japan - I'd hop up to Misawa and explore the hatchet (the whole top end of Honshu), especially Oirase Gorge (and stay at the Onsen there), Mutsu Bay (go hunting for glass balls on the coast one morning, just for the fun of it), then drive up to the ferry and catch that over to Hokkaido. A couple of days there - it's quite a different culture from southern Japan - then back south again. Hirosaki Castle is lovely in the fall, and Morioka is an interesting place. Great skiing there, by the way, if you have leave in the winter. You can take the shinkansen from there (I believe it actually goes all the way to Aomori now, but that was a development after we left, so I'm not sure) down to Osaka and then it's only a little further on to Kyoto, which is well worth a visit. Catch a hop from Yokota to Okinawa, stay in one of the beach resorts there (Okuma is awesome - go in September, after all the kids are back at school) and check out the cultural stuff there. It's nice to have the bases, and all, but if you hang out there too much, you don't really get a sense of the actual places.
You might be able to find good fares to Korea, China, Thailand, Malaysia/Singapore (in fact, you could probably get a hop to Singapore - that's how we got there), Bali (I don't know if Indonesia is on the do not travel list at the moment), even Australia/New Zealand. Europe is awesome, but in general, it's a lot more expensive to fly to Asia from the US, so you might consider taking advantage of being in the region already, and checking out some of the awesome stuff to do around there.
But if you DO end up finding a decent flight to Europe, keep us posted, tpunks have opinions about everywhere you want to go! There's some great deals through the USO, too - check the USO Japan site if you opt for an Asian adventure, or the USO Europe site to see which cities they have offices - they always have lots of tours. Also, whatever airports you pass through, check in advance to see if they have USOs, show your ID and you'll get refreshments, and a quiet place to chill while waiting for flights.