Jxi, On the Half Dome trail, there are two trails going between the first bridge and Nevada Fall. The Mist Trail is shorter and steeper, and it continues up the north side of Nevada Fall. The John Muir Trail bypasses the top of Vernal Fall and continues up to Nevada Fall. It would only take about half an hour longer to descend that way versus the steeper trail.
The switchbacks on Whitney are open, so you don't need to take the chute. But you still need crampons and at least hiking poles with snow baskets, since there is lots of snow on the upper sections.
