I am utterly smitten with Big Bend National Park and it's varied, stunning landscape. In early September I headed down to get in some quiet road cycling and exploring, since the busy season (and vehicle traffic) doesn't pick up until the October through April months. The heat can be extreme this time of year since you're on the border of Texas and Mexico and this is a desert, but I lucked out and had some cooler weather this time around.
Two nations meet here in West Texas and the mighty Rio Grande serves as the boundary. Drastic changes in elevation lend to create an out of this world, varied micro-climate experience while you pedal through your day and you're rewarded with vistas and perspectives along the way that you can only have on a bicycle.
This particular bike ride from Panther Junction's visitor center down to Castolon has become my favorite single day ride I've ever done in America and one of the top five for me in the world. That's saying something: I've lived and worked in over fifty countries on five continents designing cycling trips for a living!
If you aren't traveling with others where you can park one vehicle at the end location & then bike toward it, I'd suggest parking near Mule Ears Viewpoint. Turn right out of the parking lot onto Ross Maxwell Scenic Drive and go north to the old Maverick Rd intersection. Then turn around and go back south, all the way down to Castolon and then back up to Mule Ears. A 50 mile ride for the record books.