June is no problem for the AT, except perhaps for the whites and northward, depending on the year. June is a fine month for the rest; you could estimate your hiking speed and just pick somewhere either north or south of your destination to start. Perhaps leave the trail in southern Pennsylvania and hitch or take a bus to where you're going in Delaware.

It's not hard to find AT mileages online to work this out; the AT is a pretty easy trail to just sort of "jump on" and start walking.