Day hiking on a cross country road trip Late April.

You probably know Niagara Falls. Lots of flat land between there and mid-South Dakota. Things get interesting around the aptly-named Badlands. Picture trying to cross that with your Conestoga wagon. Hah! (Stop at Wall Drug just to see one of the country’s largest tourists traps), then on to the Black Hills. Custer State Park is worth at least a drive through (so many bison!) and Mt Rushmore of course. Devil’s Tower in NE Wyoming. (Remember Close Encounters of the Third Kind.)

Yellowstone and the Grand Tetons. I could spend the rest of my life in that area. And the Cascades in Oregon. Lots of incredible stuff along the way.

Dependent upon your route, visiting the Wallowas in Oregon or the North Cascades in Washington should be high on your list. The Wallowas are dry relative to the North Cascades, so are quite different.

When you reach the west (wet) side of the Cascades you will have entered the land of big trees. Aromas in the woods change subtly, senses recalibrate after a while.  Consequently, I suspect that if you visit the North Cascades or the Wallowas for the first time later — after your senses recalibrate — your experience will be altered. This way you have a before and after comparison available.

I should add that in late April you will probably be parking below and hiking up to the snow line anywhere in the Western Mountains. For a guide anywhere in the Rockies and further west see Snotel reports:
https://www.nrcs.usda.gov/resources/data-and-reports/snow-and-water-interactive-map

Pick a station near your intended route, and use the rule that 1 inch of snow water equivalent — SWE — translates to 2 inches of late season snow pack. It’s a poor guide unless you know the area because snow depth is so variable based on local conditions; but SWE does give a guide to the local state of melt at the altitude of that Snotel. For example, today Red Hill, North of Mt. Hood, elevation 4410, shows SWE=32.0, which is 82% 0f the 1991-2020 median for this date.

April is early for most of the mountains.  That siad, y0u should not overlook the Ruby Mountains in Nevada, if it’s anywhere near your route.

we hooked up through the upper peninsula of Michigan on our cross-country roadtrip and had lovely dayhiking in pictured rocks national lakeshore. quite a bit to do up there, just not especially mountainous