The Secret Garden Trail
You can do all this by taking the Secret Garden trail, which I named myself. I think there is no official name for this little trail, but this name of mine is ideal. Check out the pictures and you will see why. This trail is semi-marked on your topo map of the Sol Duc area: there will likely be dotted lines extended North from the Cat Walk towards Haig's Lake, and possibly to the East and South from Oyster Lake near Appleton Pass. On the flier the National Park hands out, that is the case anyhow. This trail is actually mostly worn into the hills all the way to Appleton Pass campground however. A ranger will tell you the same thing, if you seem capable of the adventure, and you don't piss that ranger off. I learned about this by being polite, and by asking to camp at Appleton Pass when Heart's Lake was unavailable. The trail is mostly intuitive, though somewhat challenging to traverse in places. You can end or begin at Appleton Pass and just remember you are going from the campground at that pass to Haig's Lake. That single sentence will ease some headaches for you. I think for that reason alone, you should travel south from Appleton Pass.
Getting to Appleton Pass
You will reach the pass by taking the Sol Duc River Trail for 4.1 miles of the prettiest, softest forest land you will ever see, and then going left/North at the fork towards the Pass. 2.6 miles of switchbacks will get you to the pass and the campground. Look around and find the right campsite for you. I spooked this bear at the East-most site at sunrise. I expect bears are more likely to the East.
Once at the Pass
Appleton Pass is beautiful, with a wonderful, sprawling campground. The only nearby water is at Oyster Lake, hardly a puddle, and for that reason, I chose to camp close in a copse of trees, but you can get as far as 1/2 mile from the campground with some gorgeous views right out your tent window. Mount Appleton is nearby, but a beast to climb, in that you'll have to bushwhack. I can't recommend that, nor could the ranger, unless you are trying to grab every summit in the Olympic Range. But you will have a wonderful sunset anyhow. (And as a note for those of you thinking: well I'm no baby, I don't mind a bushwhack- I tried it. You are going up a ridge with flowers, thorns, branches, bushes, loose rocks, bees, and everything else awful you can think of. In one hour I got about 1/4 mile. And the views were not getting any better.)
Sunrise from Appleton Pass in late summer is even better, with slow golden light tiptoeing through vast fields of wildflowers. Deer abound near Oyster Lake, which seems to be a favorite spot for them. I had 6 different deer wander past my camp, which as I mentioned was hidden in a copse of trees.
I had no tent, and only my trusty bivvy sack, so the deer were not even aware of me. From the campground and the lake, head East on the faint footpath. This is bear territory so don't stray too far into the thickets of pines.
Snowpatches will mask the trail depending on the season and traffic. Now that I am detailing this on a popular website, the trail may see more visitors, which might ease some of the disappearance problems. As the trail heads South you will notice wooded peaks ahead of you, four. These are not the popular Carrie Traverse group accessed by the Cat's Walk. These are not named on your map, and hard to even pick out as summits on smaller topos, such as that offered by National Geographic, which is what I was using. One of these 4 will be the sheerest and highest. Your trail will stay just left or East of that summit and cross the ridge at a saddle just to its left/East. Remember that and you shouldn't get lost. I'm posting plenty of pictures also.
I am not going to go into great detail about your step by step travel. You need to be able to route-find and have some topographic map-reading experience. If you need or want exact directions, than this is not the trail for you. After you traverse the several small wooded peaks you spotted from Appleton Pass, you may lose the trail for a while. Just watch for a lake. It will be Haig's Lake and aim straight for it. I did not know this when I was taking the trail, but I trusted that any lake on earth with a fish in it even once upon a time will have a foot path to it. Fishermen above all others covet solitude and secret spots.
This trail should take a few hours. Maybe a half day tops. Its pleasant and pretty. You can grab a peak or two while you are at it. I named the highest peak on the ridge "Secret Garden Peak" and the peak it shares a saddle with, which sits just East, "Wildflower Mountain". I went to the top of Wildflower Mountain to look back at the whole trail. See the picture just below. The view back to Mount Appleton and Pass was marvelous.
More details about The Secret Garden Trail
Ideal season is probably about when I was there: middle August. You should have plenty of flowers at any time in August, and most of the snow will be gone. Snow on this trail would mostly be annoying in my opinion as much of the route is "side-hilling". August is also Washington's driest month. I spent 17 days in Washington, with this trip being one of my last legs of the journey, and I got in 14 summits, and about 170 total miles, and was never rained out a single day. I did lose one morning to rain. That is much better fate than I had in Montana the same season.
About the Surrounding Area
There will be bears in this area. I saw two more before sunset when I went back towards the Secret Garden Trail for the best sunset in Olympic Park. At least, in my opinion. This spot (photos are above) is on the Cat Walk, near a single campsite which could actually hold several tents easily. You will take a spur to the left/North. I recommend it. The campsite would need to be reserved and there is water nearby, though just one little buggy snowmelt pond. Filter heavily.
High Divide Trail was a let-down after Secret Garden, and also because the sky was hazy for my exit morning. Still, Sol Duc in August is wildflower paradise. I saw plenty more and had a great morning. People in Washington do not seem to move very early in the morning, so rise with the sun, and even on crowded trails you may find yourself all alone.
Bogachiel Peak was unimpressive to say the least. But if you want to pad your summit stats, then its a short diversion of only a few minutes. The Sol Duc falls are beautiful. The scenery of people with enormous packs is great comedy. Some of the packs were larger than some of the people carrying them. Yet as usual in Washington, I was mocked for having an ice axe. Really? My pack weighed under 20 lbs, so let's not talk who has too much gear.
The total distance for the typical Heart Lake/High Divide Loop and the Secret Garden Trail extension is around 19 or 20 miles. It could be done in a day, but with scenery this lovely, I don't suggest you try. Also it can be hard to "speed" in heavy traffic and others will be moving slower than you would need to. Stretch it out.