I would say it is not necessary based on my (limited) experience.
Our sixty day window came just over two weeks ago, and I did stay up until midnight (1:00 AM Eastern actually, there are plenty of threads out there about that particularly recurring issue!). I had lists of what we wanted, backups just in case, and all the usual paranoid over-planning to which I am prone. I got everything I wanted, and then went to bed.
Over the next couple of weeks, as I am also prone to do, I fretted over our FP selections, and in particular my decision to FP Rock-n-Roller instead of Toy Story Mania on one of our days at Studios. Just this weekend I finally decided to see if I could switch out the Rock-n-Roller for Toy Story. Even though we were now at about 45 days out or so, I was able to get Toy Story for us at the exact time I wanted on the first try.
Given that Toy Story is one of the more popular rides, and I assume the FP most sought at Studios, the ability to get it at 45 days out suggests to me that waiting to the morning on day 60 will usually be okay.
All that said, I do feel I should point out the following:
1. This is only my experience, and others may have a different experience;
2. We are going during a slow period in September, and it may be different for busier periods;
3. Though Toy Story is popular, it may not be a good metric for judging how soon you need to try for things like A&E or 7DMT; and
4. Notwithstanding my reasoning that waiting until morning should be okay, I must admit that the obsessive part of me will probably have me staying up past midnight anyway next time we are planning another trip to Disney!