To save time and effort hiking, I suggest that you take short hikes on the west side (rainforest), and longer hikes on the east side (alpine).
The scenery in the rainforest doesn't change much so you can save energy by doing short nature trails or just cross-country exploring.
Your best alpine photos will be at dawn and dusk (obviously). This limits you to Hurrican Ridge, Obstruction Point, and Deer Park. You can get superior photos elsewhere, but it requires you to either camp or hike by headlamp.
(Here are a few of my favorite "off the beaten path" areas)
The best place for wildflowers is the Buckhorn Wilderness area. Species like the Piper's Bellflower (Campanula Piperii) are exclusive to the Buckhorn Mountains. There are numerous trails that lead into the wilderness, but most are long. A couple of the shorter ones are the Silver Lake Way Trail (3mi), Tunnel Creek (4.5mi), and the Upper Big Quilcine (5.5mi).
The Siver Lake trail is quickly gaining popularity; you won't be alone, but it offers great scenery for the effort. It follows a rough path beside a fast-flowing stream to a pair of lakes. From the lakes, I recommend heading up to the ridge above for amazing views of the surrounding mountains.
The first 3 miles of Tunnel Creek follow the water, and offer good photos of water cascading through a mature forest. It then heads uphill to 5050 pass, with arguably the best view of Mount Constance's east face This trail sees few people despite its quality.
The Upper Big Quilcine is notable because it provides the easiest approach to Marmot Pass. This is a good trail to get to the center of the Buckhorn Wilderness. From there you can take numerous trails following alpine ridges.
If you are willing to work a bit more for your photos, the Southeastern Olympics offer impressive sights that are rarely visited. The blueberry plants should be turning red by late August and the castilleja will still have it's color. The Putvin Trail (4mi) and the Upper Lena Lake Trail (6mi) are the best hikes in the area.
The Putvin trail is steep; you will gain over 3000 feet in 4 miles. The reward is worth the work however, as you end up at the Lake of the Angels, in the aptly named Valley of Heaven. This is a good place to see wildlife such as moutain goats and bears in late august/sptember. This is my favorite hike on the Olympic Peninsula.
While more of a backpack, Upper Lena Lake can be hiked in a (long) day. The lake is one of the largest lakes in the Olympic alpine. From the lake, a short hike takes you to the heart of the mountains.
Please ask if you have any questions.