I have had good luck at getting FP+ late in the day, even for attractions that supposedly have no more according to the
MDE app, by just constantly refreshing the screen. Last week I got an 8 PM FP+ for Test Track at 7:30 PM. Don't just give up if the app says there are no more FP+. Just keep refreshing incessantly - you'll be surprised at the times that will suddenly appear. I have obtained FP+ for nearly all of the Tier 1 attractions as fourth or subsequent FP+ by using this method, and also to get a Tier 1 FP on the day of my visit. On one of our AK days, when we went with no FP reserved in advance, I got a FP+ for Kilimanjaro Safaris for 9:15 AM while on the bus to the park that morning, even though the app at first said there weren't any. Just keep refreshing. If a time appears that you don't want, take it anyway, and then use the refresh method to modify the time until you get what you want. That seems to work even better.