You can get a personal guide but you won't avoid mobs of people in June at either upper or lower Antelope. the sun is also quite directly overhead in June creating hot spots on the walls, especially at lower. I was there two weeks ago and experienced lots of people. I've been a number of times but this was the latest in the year and I only stayed a morning.
You might try Waterholes Canyon a few miles south of Page. You get your permit at upper Antelope station, $5.00. Head west from there and go straight at the traffic light until you reach the Flagstaff Highway, about two miles. Turn left and go about a mile past the Horseshoe Bend parking area. There is a pull-off on the left. It will be just before the Waterholes Canyon bridge. Park there and walk in. It will probably have fewer people.
Best slot photography is in March to mid April for me anyway. Sun is lower in the sky allowing the glows to last longer.
Be sure and keep an eye out for thunderstorms and watch for reports at Upper and weather channel as well. Monsoon starts usually in June through July.
As alternative, you might drive to Paria contact station and get a permit for Paw Holes or other areas they might direct you to. Station is 30 miles west of Page towards Kanab.