OverviewThis is perhaps the least technical route to the summit. Certainly it involves the most amount of trail travel. All in all it is probably an hour shorter than the Davis Peak Traverse route. It will be frought with less tedium than the North Gully route. Furthermore, where as your hike up the Davis Peak Trail will no doubt be among other hikers, you can be more assured of solitude on the west side of the peak.
Getting to the TrailheadDrive to the end of Cle Elum River Road (29 miles from Rosyln, 13.3 miles from Salmon La Sac Campground) and park at Tucquala Meadow (c. 3,350 ft). There is a campground here and two trails. You want the one to Cathedral Rock (standard route to Mt. Daniel). It crosses the river on a good bridge.
Trail to Lake MichaelOn the other side of the river look for a sign saying something to the effect of "to Lake Michael." Find the sign. Search for the trail. More on that later. Continue up toward Squaw Lake (4,841 ft). Suffice it to say the trail makes more turns and switchbacks than is shown on the map. At the last southerly switchback at approximately 4,440 ft go 10-20 yards beyond (north) of the switchback to where the trail goes over a manmade rock culvert. On the left here is a stream indentation. I think this may actually be the old Trail Creek Trail, the one that goes toward Lake Michael. Now there may be a new trail farther north toward Squaw Lake but I didn't go that far to find out. But that will be out of the way.
At the rock culvert leave the trail and go about 40-60 yards into the woods perpendicular to the trail. You should pick up the old trail. If not, simply continue bearing SW to the 4,500-ft saddle and marshy Squitch Lake. The Trail Creek Trail makes a tangent with the north side of the meadow. It is easily followed and later "junctions" with a better trail (may be the new trail from/to Squaw Lake) after a slight downhill section. Once on the nicer trail take it for another 45 minutes or so to the junction with the Lake Michael Trail (c. 4,250 ft) on the left.
Hike the Lake Michael Trail (initially steep but later mostly flat) for maybe 3 miles to Lake Michael (5,100 ft). In actuality, you don't want to go all the way to Lake Michael (rimmed by a headwall). Instead, at 5,000 ft a few hundred yards before the lake, you will have to leave the trail and bear SE up a side valley to get to Goat's summit. Allow 3 hours to get from the car to near the lake.
From Trail to SummitLeave the trail just before the lake and climb up the north side of the stream coming down from the minor, narrow basin northwest of the summit. There are some low angle slabs (Class 2/3) with interstitial brush to negotiate to get into the basin, but once you do the going gets easier. Climb all the way up to the headwall below the final saddle west of the summit. There are several steep Class 3 or Class 4 options to gain the saddle. If you go rightward (westward) you will have to drop around to the south side of the crags that make up the center of the saddle. In the middle a timbered spur reaches high up to near the center of the craggy saddle headwall. At the apex of this spur where it abuts the wall move left and up to a white snag. Climb over or under the snag then walk a horizontal ledge leftward (east) to its end then up and right to the craggy crest. Easily walk eastward off the rocks from here to easier ground. Class 3. Note that there may very well be easier routes. This is the way I came down and I found it quite doable.
Once on the saddle, simply walk up the semi-timbered final slope to the summit, which features a flat top, a cairn, and a makeshift register in a pipe.
Allow 2 hours from the Lake Michael Trail and 5 hours from the car. Could be done in an hour less than this if speedy. The gain is roughly 4,000 ft adding in the various ups and downs. Distance is maybe 9 miles (8 of which are on trail).