I think there are some daylilies that should bloom in June. Mine didn't start blooming until late June, but I know there were people posting pictures of their daylilies before then in the Daylily forum. Maybe go back through the late May / early June postings in the daylily forum to see if you like any of the particular cultivars that were shown as blooming. That's if you like daylilies of course ;)

This will be almost completely related to your specific zone as things bloom earlier in the warmer zones, later in lower zones, and is somewhat variable in any location from year to year. Look around your area at other people's gardens for the most accurate info, and check the stores.

I still have some notes from when I lived in 5b OH, you might consider...
Aquilegia
Bergenia
Daisy (not sure which ones I had)
coneflower
daylily
Hosta
Monarda
Phlox paniculata
various roses
Scabiosa
Silene
Stokesia
Tradescantia
Linum (blue flax)
Heuchera
Dianthus
Cerastium (snow in summer)
Agastache

If the foliage of the plants you choose is interesting and varied, tons of flowers aren't necessary for it to look great at any given time.

You may also get a lot of bang for your buck by installing some annuals a few weeks before the big shin-dig. A couple of attractive hanging baskets can make a huge impact.

Thanks!

A few things that were flowering in june in my zone 5 garden:

Late peonies -- Sarah Bernhardt and Kansas were both blooming mid June this year although that was even later than usual.

Clematis Mayleen and Vyvynn Pernell

Allium purple sensation

Annual larkspur--sow seeds in fall
Annual nigella

Biennials sweet William and campanula Canterbury bells

Pink chives
Baptista
Amsonia
Early bearded iris
Bleeding heart was still flowering in part shade

Thanks for the great list!

I've bought those allium bulbs to plant this fall, and it's good to know that bellflowers are early, I planted one this year and I'm looking for a couple other varieties. I love amsonia too, I just need to find a spot for it lol. :)