Stretch your definition of a day hike with a trip to Shoe Lake. This gorgeous lake is the perfect introduction to the Goat Rocks wilderness, but because it's closed to camping, you have to hike even further into the backcountry to overnight. The way is gently- to moderately-graded, making for relatively fast hiking, and the views along the way are a perfect unveiling of this gorgeous area: first deep green woods give way to an airier, fragrant forest, then after an alpine traverse you burst onto the Shoe Lake basin and stunning views of the Goat Rocks beyond.
Start from the PCT south trailhead, found a quarter-mile east of White Pass, just inside the Gifford Pinchot National Forest boundary. This woody trailhead can be dense with mosquitos, but once on trail you should have smoother sailing. There's not a lot of room for cars, maybe six or so can fit without blocking the roundabout, which is important to keep clear for horse trailers.
Fill out a Goat Rocks wilderness permit (they're free, just write fast so the bugs don't get you) and set off down the trail. It climbs through a dark green forest with a high canopy for about 2.2 miles, passing a talus field where you may hear squeaking pikas in the summer. At 2.2 miles cross into the Goat Rocks wilderness boundary at Ginnette Lake, a nice enough little body of water but in early summer rife with mosquitos.
Keep on truckin', hiking another 0.2 miles to a nondescript junction. To the left is a trail to access Round Mountain via Twin Peaks. Stay straight on and pass Hell Lake, hiking one more mile to another small junction with the Chairlift Trail 1112. This half-mile trail leads into the White Pass Ski Area, but it's a flat, buggy mess and not worth the side trip.
Continue another half mile before the views really begin to improve here. The trail begins to climb in earnest, leaving the wilderness area and tracing the border between the forest and the ski area. Interestingly, these are some of the best views you'll enjoy so far -- look east towards Rimrock Lake and south at Hogsback Mountain.
1.7 miles from the junction with the Chairlift Trail, pop through a small saddle and look at the trail traversing this high hillside in front of you. You can see the full 1.3 miles you have to go to the pass above Shoe Lake, and if there are others sharing the trail, you can follow their progress and see where you need to go.
In early summer, this traverse can retain snow late into the season. Be sure to have some sort of traction devices in order to cross safely; the runout on the snow slopes is steep and it's a long way to little Miriam Lake at the bottom of the basin.
Once you're safely across, stop for a moment and enjoy the views from the pass above Shoe Lake. You can get the full experience here, because trails encircle the lake. Head down the trail into the basin and at a large tree, an unmarked junction splits the trail.
Go left for a hiker-only path that's 1.3 miles to the lakeshore (remember, there's no camping here, but you can certainly lounge in the sun for the afternoon if you wish).
To your right, the Pacific Crest Trail (PCT) continues around the backside of a small high point before switchbacking and reentering the lake basin 1.1 miles from the pass. You can then hop on the hiker-only trail and do a full loop of the lake by climbing back up to the pass before heading back the way you came.