Listen to Tuvalu. She knows everything. As soon as you tap your band for your last FP, you can look for another one on the app.
It's luck of the draw, but try to pre-book the most desirable FPs and then ride the less desirable rides on additional FPs. At MK last week, we rode pretty much everything on FP. Many of the less desirable FPs were available late into the afternoon and evening. On Thursday, we didn't even get to the park until 11:30, used our 3 FPs before the 3 pm parade, and then managed 5 additional FPs plus a TS dinner and still left the park by 7:30. However, be aware that doing the park this way may result in a lot of walking as you criss-cross the park from ride to ride if you don't choose rides close to one another.
If you Google, there are a couple of websites with lists of which rides to prioritize for FP and which rides usually have FPs available later in the day.
We had less success with additional FPs at other parks because there are fewer options but we also had other plans than rides at those parks (shows at HS, World Showcase at Epcot, etc.).