Hi Jammer, You would need an overnight permit of some sort to spend overnights anywhere in the area, whether it be on the east side of Whitney, in Inyo National Forest, or on the west side in Sequoia National Park. So the Day Hike permit would not suffice.
All the agencies, National Forest and National Parks, require that you have an overnight wilderness permit, but it doesn't matter which agency issued the permit. Even PCT hikers starting from the far south (or north) need to carry a permit issued by the PCTA organization.
Last year, there were surprisingly few last-minute online permits available for overnight hikes. I'm pretty sure the software has not been built to automatically release the no-shows into the online system. I think last year the only slots showing up online were the group-size reductions occurring when people were picking up their reserved permits. I am sure hopeful that the no-shows will be made available this year.