I think a good benchmark for the unsupported record would be to hike to Pearisburg in 25 days, which is an average of 25 miles per day. If you can reach Pearisburg in 25 days, you have a shot...go back to Springer and start north.

The jump from 25-36 would not be nearly as bad as the jump from training to 25 per day in my view, plus you would get your trail quirkks worked out, like blisters, shin-splints, etc...take a week off and do some light maintenence training and go for it! Furthermore, the run to Pearisburg would prepare you for this feat, you might find after averaging 25/day you are no longer interested in going for Ward's record.

Good luck, a guy called Gailee Man from Israel came very close recently on his first hike, very close, he has a trail journal you can read.