If you're looking for a meeting place or a good stop around ½ or halfway between New Middletown and Davenport, FL, you can try Charlotte, NC, which is about 7½ hours south of New Middletown, or about 9 hours north of Davenport, FL.
Other possible places to stop near the midpoint include Rock Hill, Springdell, Lancaster, White Oak, and Columbia.
About ¼ or a fourth of the way, you can stop at Beckley, WV. This is about 4½ hours south of New Middletown.
About ⅓ or a third of the way, you can stop at Bluefield, WV. This is about 5 hours south of New Middletown.
About ⅔ or two thirds of the way, you can stop at Savannah, GA. This is about 11½ hours south of New Middletown, or about 5 hours north of Davenport, FL.
About ¾ or three fourths of the way, you can stop at Hinesville, GA. This is about 12 hours south of New Middletown, or about 4½ hours north of Davenport, FL.
If you drive straight through without stopping at all, the total distance is 1,050 miles. The distances shown above don't include local roads to get into each town, so if you're planning a long road trip, make sure you factor in traffic and road conditions, rest stop breaks, and interesting detours along the way. If you're in a rush, you can do the trip with zero stops, and your total travel time would be around 16 hours, 35 minutes.