First, you need to get to Clayton, New Mexico. Travel time to Clayton is not included in your two hours.
From Clayton, you're going to drive northeast on US 56 to Cimarron County, Oklahoma.
That's right, Cimarron County. The least populous and least densely populated county in Oklahoma, it is the western tip of the Oklahoma panhandle. Here's the layout.
As soon as you get to the New Mexico/Oklahoma state line -- where US 56 enters the map at the lower left -- turn south on a gravel road and drive about a mile and a half. This road will end at a T intersection, where you'll see this.
The post behind the sign is the survey marker showing the point where the northern border of Texas meets the border between New Mexico and Oklahoma.*
You can get to the post via a nearby cattle guard, and you do the obvious thing.
Voila! You have now been in New Mexico, Oklahoma, and Texas, and you're only about 15 minutes in.
Return to the highway and continue northeast to the nifty roundabout at the heart of the county seat, Boise City.
Pretend you're heading to Guymon, but you're not; you'll actually turn on the edge of town to keep going northeast on US 56. Allow yourself to be hypnotized by the minimalist charm of the Panhandle landscape. When you see the sign marking the Texas County (Oklahoma) line, at the upper right hand side of the above map, wake suddenly from your trance and careen left onto another gravel road.
This time, you're going to drive about five miles north, but it's a good road so no worries. You'll reach another T intersection, and... nothing. But turn left, and drive another mile or so, and suddenly you'll see this up ahead on the right:
Awesome. You'll run up and inspect it, and stand inside of its metal frame (probably an old windmill) so you can be in three states at once. But then you'll notice there's no survey marker, and after you putter around for a while you can find this back in the road, if you didn't park on top of it.
...and once again, you do the obvious thing.
And there you have it! You have now been in Texas, New Mexico, Oklahoma, Colorado, and Kansas, and it is not even lunchtime. Turning around and heading straight east will get you back to the highway at Elkhart, Kansas. From there, you're on your own.
Mrs.5000 and I (and our intrepid travel mascot, obviously) did exactly this yesterday morning as part of a really lovely expedition on the Great Plains. You'll be seeing bits and pieces from our fieldwork for quite a while.
* "But New Mexico is WEST of Texas, not NORTH of Texas!" Good eye. However, there is a jog in New Mexico's eastern border that makes the state extend two miles further east in this area, into space you'd expect to be part of Oklahoma. If for some odd reason you want to stand on the northwest tip of the Texas Panhandle, I can tell you how to do that too, but at that point Texas shares both its western AND northern borders with New Mexico.