Take the I-10 route through Houston. I have done it many times. Went through west bound last Monday at 11:00 and traffic was not an issue. Don't know if there is such a thing as no rush hour. I drive through at around 50 MPH and traffic seems to flow better for me. Be aware of lane changes and move over early. Some one gets in front of you, slow down a little more and give yourself a little extra room. All of the loops are longer and I find them to take as much or more time to get through. Traffic can be just as heavy at tines,
45 south of Houston traffic can be just as heavy as taking a downtown route via I-10. The 610 loop has heavier traffic than the north side and has some very rough sections.
If going to Big Bend you have to make a trip to Ft. Davis and McDonald Observatory. Stay at Ft. Bend state park. While there take the switchback to the top. From there you can get phone signals. Large sights in the valley. Has Wi-FI and cable.