Snow will be thinning out and it is likely the trail and the switchbacks will be accessible by then. There may still be some icy spots, but that doesn't bother most people.
However, if you want to be cautious, try again on May 1 when a number of non-confirmed lottery reservations are dumped on the system (at random times throughout the day), and try for a later date.
I am sure walk-ins will be available all summer, but parking is the problem. This year due to road construction, people cannot park along the road, and only the parking lot will be available. Therefore, Inyo N.F. will be giving parking permits ONLY to those holding reservations. If you can get someone to drive you up to the trail head without needing to park, then the walk-in permits are good to go.