Arkansas has great intro-to-mountain hiking. http://www.petitjeanstatepark.com
You can get a RT flight to Atlanta for ~ $200, stay at a hostel jump off point for around $100, transportation included, then practice hike right on the AT! Check out the Hiker Hostel in Georgia, although it's recently sold. Hike Inn is amazing
You could always do the most beautiful hike in the world: the John Muir Trail. It's a little over 200 miles spanning from Mt. Whitney to Yosemite. Permits can be tough to get, but lots of folks have hammocked the whole way if you plan your stops below treeline. The elevation and planning/resupply can be a challenge, because it's a high and tough trail for most people. Usually takes about three weeks, and is pleasant from June to September (with some earlier times being extremely buggy).
There is also the Long Trail in VT. It's also about 250 miles, and full of good trees and backcountry camping.
Any given section of the Pacific Northwest Trail... I'm partial to the section from the ID/MT border to Oroville, WA as it includes some of the most wild areas left in the lower 48 and the valley I grew up in. Near guaranteed solitude, last time I did the kettle range (3-4 days worth of trail) and only came across one other hiker. That whole section is almost exclusively national forest with no hammock restrictions at all. You could also extend it east with 30 days to work with and get some Glacier NP in there too
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