Where to stay on our trip South.

(I note that there are two Newports in Oregon.  I have assumed that your Newport is the inland one.  The map pins in this program assume that it is the one on the coast.  If it is the coastal one then my routes are all completely wrong.)
Going that way, I can't figure out any even segments - either you are only driving 3 hours or it is 5+ hours.  So Newport to Umatilla is about 3.5 hours but then going to Jackpot is 6.5 hours.  I don't know that area well enough to know where you could stay that was 2 hours past Umatilla.  Then Jackpot to Ely (Nevada) is only 3 hours, and Las Vegas is another 4 hours.  Parker (Arizona) is another 3 hours past that.
There is an eastern route on the map which goes to Butte, MT, Pocatello, Salt Lake City and through Saint George to Parker and that is 19+ hours and appears to be more populated.  
The western route takes I-84 for the first part, but instead of going to Jackpot, it takes US 95 down to McDermitt and then is both on I-80 and US 95 for a bit.  Then it follows US 95 down to Fallon and Barstow before taking I-40 east and US 95 down to Parker.