don't know how it will be in June for you, but in late May, we did the evening EMH at AK (after a 5:30 dinner at Tusker House) It was fairly busy until about 8:00 p.m. on a night when EMH was 7-10 . But even 7-8:00 was not as busy as AK gets during the day. And after 8, it was empty. DH & I sat on a bench just off the area where the Dino rides are...we'd see someone walk by about once every 5-10 minutes, and there were no lines for the Dino rides like triceratops spin and whatever that other one is called. That's how empty it was. DD & her friend went to ride Exped. Everest -- they rode several times in a row with no waiting. And it's fun to ride in the dark -- you can see so far from the top, and all the lights twinkling...cool!
I think what happens with AK is most people want to do the animal-related attractions, so they're at the park all day. And it's hot and very crowded...by 4:00 p.m., they're mostly ready to leave, and not wanting to come back after dinner. So it's a great park to go to at night if you want to do just the rides (cuz all the animals have gone beddy-bye.)