That permit availability page is updated only every few days. For example, today is June 21 and the page was last updated on June 14. So you cannot depend on it for timely alerts.
I never reserve a permit in advance, and have never failed to get one when I walked in. This includes the middle of summer on weekends. Last year in late summer / early fall they told me they had closed every day that season with main trail openings left. You just have to know the special times when your chances are best. See step #6 under Feature Topics – Wilderness Permit Options – in the left column.
Overnight permits are easiest. Just walk in on the day of entry shortly after 11:00, when all the no-shows are declared SOL. You will still have plenty of time to hike to Trail Camp by dinnertime.
For day permits, your best bet is to show up shortly after 11:00 the day before and hang around. Someone with a reservation, but with fewer people than planned for, will soon show up. Get one of them. Enjoy the Alabama Hills, grab a burger dinner at the WPS, bed down at the Portal until the time you choose to start up the next day.
There is more to the story that a little research and inquiry can uncover, but these are the important points.