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I went last year. In May. It was 108 degrees F.
You are going to roast OP. Bring light weight long sleeved shirts, a hat with a big brim, a big canteen and sunblock. Water is available throughout the park. The water in your Camelback will need to be changed out a few times a day because it will get hot.
Hike in the mornings (hanging rock) and night. (I assume you are camping in the mountain campground, it is the best) There is a trail you can hike at night from the chisos mountain campground up to a rock out crop and see the brightest stars you will ever see in your life. There are so many, you will have a hard time picking out the constellations. No lie.
Spend the hot ass afternoons in the Santa Elena Canyon under the shade of the only trees in the park. Go rafting on the Rio Grande if there is any water left this type of year. You will find the water is really low in the summer.
The boat that crosses to Mexico is gone. The border is closed in the park. The closest border crossing is in El Paso now.
Pic related, its my bike entering Big Bend