Lots of public land with over the counter elk tags available in Colorado if you hunt 2nd, 3rd, or 4th season. It can get crowded and success rates are low in those areas. The prime areas will be draw only as is most of the state during the 1st season. And it takes several years to get enough points to get picked.
You can hunt 3-4 hours drive from Rocky Mt National Park. I'd arrive a few days before I planned to hunt and visit the park 1st. Then move on to where I planned to hunt. The weather is usually decent 2nd season. By 3rd the chances of snow and passes being closed increases. By 4th season the weather can get pretty bad. But you'll see fewer hunters 3rd and 4th seasons.
My wife and I have made multiple summer camping trips to Colorado, Yellowstone, Grand Canyon and Utah in the past. I'd hunted there a few years earlier, but my wife had never been in the west during the fall. We drove to Colorado to elk hunt in November 2018.
We left early enough to take our time driving out and see some places we'd never been. Arrived 4-5 days before the season to scout and visit local sights. It was a combo hunting trip for me and vacation for my wife. After the hunt we took the scenic route home. From home to where I hunted is about 24 hours driving time. I have pushed it into 24 hours straight by swapping drivers and sleeping, but usually spread it over 2 days. This time we took 3 1/2 days each way with several detours to see stuff. Even though I didn't kill anything it was a great trip. Plan to go back this fall.