Go for it tomorrow!!! Today's experience gave you a bit of acclimatization.
Not sure how early you plan to go, but if you have more time this evening, hanging out at Whitney Portal is far better than sitting down in Lone Pine. Every bit helps!!
If you come back another year, drive up to Horseshoe Meadows. Spend an overnight if at all possible up there. Do a short hike up there, too. Trail Peak makes an afternoon easy hike. The altitude at HM is about the same as Outpost Camp, and it accelerates your acclimatization. It's a 45 minute drive from Lone Pine.
