Not sure how fast you wanted to travel. Do you want to park somewhere for 7-10 days and explore within 50 miles, and then move 100-200 miles? Do you want more leisurely?
If you want a May 1 to Dec 15 itenerary and this is to be a one time East Coast visit, what I would suggest is different than a summer in just the south and head to Fl.
If this is just the summer in the south, I would suggest leave La, head up to Nashville area, see a couple of the state parks, continue to mammoth caves, and then east to the Smokey Mountains national park by mid june, Gatlinburg, Dollywood, Boone N.C. then up to West Va by August, Hot Springs/ Natural Bridge, keeping to higher elevations until after labor day and visit Shenandoah valley, Harpers Ferry, down Skyline Dr to Blue Ridge Pkwy Stop for a day or two in Galax and at Fairey Stone state park, then down to Mt Airy and Boone NC, Blowing Rock area is beautiful, Mount Mitchell if you can. and then run east to the Outer Banks, Hatteras Island around late September, down to SC state parks around Charleston, then On to Savannah Ga. See the old cemetary, spanish moss in the trees is awesome and many of the older tombstones tell the life stories of the person interred. If you want a city swing west to Atlanta, Aquarium there is awesome, Athens mansions are impressive examples of late 1800's style construction. And by now November should be ending, so go to Fl and snooze,