I'm going to do a couple sections of the Ozark Trail in Missouri over spring break. Its not totally finished yet, but about 200 miles are complete so far. It will eventually connect to the Ozark Highland Trail for a total of 700 miles.
There shouldn't be any snow and you'll probably encounter very few people. Resupply isn't really an option though, as it passes through some pretty remote places. The sections of the trail I have done are great, plus I live less than an hour from the northern terminus.