I not sure about what you mean by not driving long hours, unless you plan to fly or ride the greyhound driving long hours is the only way. We were late getting off and had resevations in Yosemite. We left North Florida and were in California (Panamint Valley) in 4 days and in Yosemite in 4 1/2 days.
We did it by being on the road at 7 AM at the latest, minimizing stops, no long lunch breaks, and driving until 8 PM. You gain an hour driving west for each time zone so we were on the road 14 hours a day. Eat in the car. We were pulling a popup and so we couldn't drive 75 mph.
Day 1 left N FL, made it to east of Dallas on I-20.
Day 2 Made it to just across the NM border
Day 3 Kingman AZ
Day 4 Panamint Valley CA
There is no other way.
You need to pick an expeditious and as close to direct route as you can find too. No following the Interstates and you cannot afford to be caught on minor highways with lots of stop lights either. We were familiar with the route ahead of time.
For us there are direct routes east to west. For you cutting diagonally across the country it will be much harder.
You will arrive exhausted.