I'm really confused, when you initially posted you talked about driving just 5-6 hours a day, now you're saying your in Kentucky and planning to get to Omaha next - which is 10-11 hours away and the upper limit of how far you should travel in a day - and that's assuming you're at the Kentucky/Illinois border.
Then you're talking about going from Omaha to Billings, which is about 850 miles and about 15 hours on the road, and nearly 50% more than you can safely travel in a day.
So beyond the obvious that you need to stop well before you get to Billings, you also need to rethink your definition of mountains. It really isn't a case that you go up a big hill and then come down and you're done - you'll be going over a few mountain passes, but also several more smaller hills, starting when you go through the Black Hills (which are actually mountains) in South Dakota.
Billings is a perfectly good place to start before you head into the most significant mountains of Montana, but you'll need to stop someplace before that - perhaps Rapid City or Wall SD.
Similarly, if you stop in Billings, you'll need to stop again somewhere again before Seattle - Coeur d'Alene, ID could be an option.