That would be an absolute minimum with no stops.
Based on a best-guess fuel economy of 35 mpg, each tank would cover 245 miles, which means his 2829 mile route required 12 tanks of gas. If you account for an average of 30 minutes per gas stop and substract 11 enroute gas stops (5:30) from his overall time (38:49), you end up with the 2829 miles being covered in 33:19. This yields an average cruising speed of 85 mph.
When I have to slab-blast across the country, I've noticed that I typically average about 75-80% of my highway cruising speed (factoring in traffic, construction, speed traps, etc.). If you apply this logic to Reese's ride, it follows that he probably trying to maintain a cruising speed of at least 105mph and going faster (115+ mph) whenever conditions allowed for it.
Based on the photos, it looks like he had about a 3 gal auxiliary gas tank, which would decrease the number of gas stops to eight (8) and reduced the raw average speed to 81.2 mph. That infers a maintained cruising speed between 101-108 mph.
Either way, one heck of an accomplishment.