Welcome to the Northwest, I hope you find it to your liking.
I believe June - October is fair estimation of traditional hiking season in western Washington, although this year July was pretty snowed in after a wet and cold spring. June at elevation is usually pushing it early in the month on the west side of the Cascades, at least in the Central Cascades north.
Off the top of my head, others I would add to your list (and by no means are my suggestions definitive. My experience in mostly in Washington. Others here will comment on the wonders of Oregon . )
So, in no particular order I give you a list:
1. Spider Meadow - Spider Gap - Lyman Lakes - Cloudy Pass - Image Lake (or many alternatives such as Miners Ridge) - Buck Creek Pass. A classic loop. Lovely. Outstanding views.
2. North Fork Saulk River / Pilot Ridge / White Pass / White Chuck River - a great looop with terrific ridge hiking and tremendous old growth.
3. Pacific Crest Trail - Snoqualmie to Stevens Pass (quite nice), Stevens Pass to Border (outstanding). The northern parts of the trail venture through the Henry M. Jackson and Glacier Peak Wildernesses, the Pasayten Wilderness and North Cascades National Park. In other words, an epic treat. A grand introduction to the wild parts of the state.
4. Goat Rocks Loop - a great three day trip - or for a longer stretch, you could hike Pacific Crest Trail from Mt. Adams over the Goat Rocks into Packwood. This hike works in excellent weather - the views and knife ridge is what sells this hike.
5. Enchanted Valley - Anderson Pass - La Crosse Pass - Hart Lake - O'Neill Pass - Great loop through some of the prettiest territory in the Olympic National Park.
6. Cascade Pass / Sahale Arm trail – An inconic hike. Do as a dayhike or hike up the arm and then go back down to Cascade Pass and hike the 25 miles to the town of Steheiken (only accessible via boat, float plane or on foot) on Lake Chelan.
7. Ptarmigan Ridge - Mt. Baker (near Bellingham) - Short hike relatively, but some fine camping spots and really, the views-per-mile, especially in fall, is outstanding. 5-star dayhike.
8. Olympic National Park beaches - I very much enjoyed going out to Forks and heading north of along the beach. Warning, slow going and you need to pay attention to tide charts. Great spring / fall hike.
9. West Coast Trail, British Columbia – I haven’t yet done this one but plenty of people here have – a terrific coastal trail with many challenges.
Spring fever bonus! And just because you will go nuts waiting for hiking season to start in spring, go over to the Black Canyon near Yakima. Easy hiking up to beautiful ridge views of Yakima Valley and beyond. Can be done as a dayhike or overnighter.
Finally, I would encourage you to visit two websites. Become a member of The Washington Trails Association and Northwest Hikers Forum. Both have many great resources and suggestions! Welcome!