I can't remember if I had to refresh each page once it was midnight, but if I did, I didn't have to log back in or anything. I selected our time first, then once confirmed, clicked on each page after that to select the same time. Once I had everyone's time selected and confirmed, I went back to the first page and closed them all out.
We had our choice of times beginning at 10 am, but decided on 11:00-11:30, and that worked out perfectly. We arrived right at 11, dropped our luggage, and literally walked past everyone who didn't have that check in time. We never stopped walking until we got to the check in desk. Those in our party who weren't platinum had about a 10-15 minute wait before they boarded. This was in Galveston.
Also, it's midnight EST. We're in the mountain time zone, so we were able to check in at 10 pm, MST.