At the false summit you'll be staring at the true test summit of Goat Mtn which is about 40ft higher. Go back down a bit and cross the permanent snowfield. There is a clear scramble path up through a rock gully and through scrubby trees.
Entrance to Nisqually Chutes is off the Muir Snowfield at approximately 8200ft. Start looking for the entrance at 8500 ft and the slope is obvious as it empties out on the Nisqually Glacier where there is a clear open slope to the moraine.