I summitted Longs Peak in Colorado (14,259ft) on a warm, sunny Labor Day. Short sleeves were fine up until about 12.5k, after which a long-sleeve flannel shirt felt good. In the shade, I added a warm insulated hoodie. Several layers are better than one heavy coat.
Examine your route; if it's facing the sun, you will stay much warmer than if you have to climb through the shade when you're nearing the top. Take insulated gloves!
I'm not sure if conditions are similar at Huron, but at Longs it was extremely important to begin climbing long before dawn; thunderstorms tended to come in early afternoon, so leaving the summit in time to get below the treeline before lightning started striking was important!
It may be colder in June than my September climb was, so do your research about if you're taking enough clothes!
Edit: Coming from a fellow flatlander, do everything you can to get acclimated before you make the climb: go jogging in camp, kayaking, biking, whatever you can do. You'll appreciate it on the way up. Had I not already been acclimated, Longs would have kicked my butt, and I may not have made the summit. Even when you are acclimated, there's a world of difference between the oxygen available at 12kft and 14kft! This has a lot of bearing on your stamina.