Black cohosh (Actaea racemosa) has a white spike of flowers about then. Goat's beard (Aruncus dioicus)is another plant to consider, but also white.

Monarda fistulosa is usually a pale purple bloom and it grows well in part shade. Later in the year, Helianthus divaricatus might be an option for yellow flowers in the sunnier areas of the woodland.

What about lupines...aren`t they traditionally found in the sunnier areas of the woods

Also be sure to visit the parks and nature trails around you. See what may be blooming along those that you like the looks of. If you don't know what they are, take some good pictures of the flowers and the leaves (incl. leaf arrangement) and ask for an identification on the Name that Plant forum.

FataMorgana

What about bleeding hearts? Mine is still blooming in June and the fern bleeding heart blooms all summer.

Heucheras
2 biennials: Sweet William (Dianthus barbatus) & forget-me-not (Myosotis)
Corydalis lutea
groundcovers: Galium odoratun (Sweet Woodruff) Lamium mac... (spotted dead nettle),& Ajuga reptans
perennial cornflower (Centaurea montana)
hardy geraniums G. phaeum & G. macrorrhizum (bigroot)
Polemonium caeruleum (Jacob's ladder)
Sambucus nigra (black elderberry)
Columbine

This year our spring & early summer was cooler than normal so blooms were later so might not match up with your season. I'm in western WA zone 7, so timing might be different for you.

Virginia Bluebells I think.

Disporum flavens , Glaucidium palmatum, Smilacina racemosa, Smilacina japonica, Lilium martagon(June/July), Trilliums, Polygonatum, Primula japonica(moist), Gillenia trifoliata, Trollius(moist)