Hartford and New Orleans are about 1,267 miles apart by road. Split the difference and you arrive near Charlotte, the city closest to the halfway point between them. That puts roughly 13 hr 14 min of driving on the Hartford side and 13 hr 36 min on the New Orleans side — the fairest single meeting point among the cities near the middle.
Over this distance most people will fly rather than drive the whole way. Charlotte still makes a fair, central place for Hartford and New Orleans to converge, splitting the travel instead of asking one side to cross the country.
If Charlotte doesn't have what you're after, Raleigh and Atlanta are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Hartford and New Orleans.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.