Grand Canyon in late May / early June is going to be very hot inside the canyon – a companion almost passed out from heat exhaustion while hiking up from a river trip in mid-May years ago. Also, backpacking permits on well-known trails go fast and early, might already be too late.
That time of year you’ll want to be in higher elevations unless you know how to not get baked in the desert.
4 hours drive from Las Vegas airport can (barely) get you to the east side of the Sierra Nevada – Mt Whitney approaches, PCT, JMT, and many other hikes; 3-4 hours to the mountains around Los Angeles – PCT, Mount San Gorgonio, Mount San Jacinto (take the Palm Springs Tram up & down); 3-4 hours to southern Utah including Zion, Bryce, etc.; 3 hours to Telescope Peak in Death Valley; 3 hours to the White Mountains in CA/NV if you are very adventurous. Probably a bunch of nice stuff in northern Arizona including the northern AZT, but I don’t know much about that area.
If I had to choose one, and could get permits, I’d go for the eastern Sierra.
— Rex