SE: Those spots have never been reported as full at all, so yes you should be able to find a spot. But...
The big problem is parking. You will get a parking pass when you pick up your day hike permit, but I think it is only valid for the day of your hike. They're issuing parking passes, one per party, this year, due to the road construction and no parking allowed along Whitney Portal road for any overflow.
So I don't know about the parking on Saturday night before you start hiking. You should call the wilderness permit office (760-873-2483 between 8 am and 4:30 pm) and ask about parking the night before -- see if what you want to do can be accommodated.
...and please, post here what you find out.