We try to use our FPs early in the day and search the system for the 4th+ FPs sooner in the day. We also like having our 1st FP be in the first hour window so that if we oversleep ropedrop we still have a ride to walk-on, and if we make ropedrop we do those rides before hitting our first FP at the end of its window.
Our trip this past March was onsite for 5 days and we were not able to secure a FoP FP until the 5th and last day of our trip. After we used up our 3 FPs at AK that morning (starting with a FoP FP), I started refreshing the system again while we were eating lunch around 1pm and found another FP for FoP at 2:30p. We've had much luck with this method over our trips for our party of 3. We even found an Illuminations late on our Epcot day too.
We grab any FP that seems good, then keep going back in to modify either for a better/closer time or ride. At least you'll have something while on the prowl for better. Goodluck! Just because a ride shows no FPs left for that day does not mean some will not show up later, usually because other people release them but sometimes WDW adds extra in too.
Refresh is your friend. The easiest refresh is by just clicking the time windows at the top. If the system is giving you trouble, close the app and re-open. If the app is down, try using safari or the browser on your phone. Once you reach the tapstyle for your FP ride, look for your next FP right there on line (if pressed for time, grab any FP and modify later). MK usually has the best availability, we often get 8+ FPs/day without even trying too hard during average crowds.