Coming to the USA for a somewhat random, non-climbing specific road trip later this year and would like to get a bit of climbing in at as many classic areas as possible along the way. Our route takes us across the southern USA in late May/early June. Is a day or two in Hueco possible that time of year? Never climbed in the SW or at Hueco specifically, but even a couple of hours on some easy classics would be a dream come true.
It's highly likely to be unbearably hot at that time.
I've always thought that if you're a desert rat you might be able to chase shade in the morning and have a good time during summer. On the plus side it should be easy to get in (which is not the case in winter/spring, owing to the limited number of entries allowed).
You're likely to be perfectly fine. The windy season should be over (that's usually April or so). You'll start around when the park opens (still 7am, I believe), and your sessions will end noon or 1pm. Carry more snacks and water than you think you need, and rest more often. Hire a guide if you can, East Mountain has some really great stuff. Make sure and get burritos at Rafa's or tamales as DoÃ±a Lupita's.