After you finish your initial 3 fastpasses, you can go to a kiosk at another park to get your 4th fastpass (It can be any park you just have to go to the kiosk IN that park). The fastpass kiosk only works for the park you're in. After you use your 4th, you can get a 5th, and so on...
This is exactly how it worked for us as well when we went in March.
Unfortunately, I found that most of the time, by the time you got to the second park, most of the worthwhile FastPasses were already taken (and I wasn't going to get a FastPass to skip the 5 minute line for Living with the Land just because the option was available). We did have some REALLY good success at Magic Kingdom one night scoring FastPasses to Jungle Cruise (which had a 45 min stand by at the time), Winnie the Pooh (which had a 30 minute stand by and my wife had never been on it), Haunted Mansion (which had a 30 minute stand by at the time) and Space Mountain (which had a 60 minute stand-by at the time) all after having three reservations in the morning at Animal Kingdom.
Just be smart with your "extra" passes. You can only have one "extra" pass at a time and if you've used up your first three by lunch time, depending on your priorities, you may want to avoid booking a 4th for something that has a very late entry time (no matter how good an option) if a decently good option is available substantially sooner. Also, be careful you don't fall into the trap of waiting in long lines at the FastPass kiosk. If you wait 20 minutes at the kiosk (which does sometime happen) only to get a FastPass for a ride that only has a 20 minutes posted stand-by line, you really didn't save any time.
We would typically use the MDE app in advance to see wait times, narrow down our choice to two or three attractions that we wanted, look for a kiosk with little/no line (typically while on our way to another attraction), then select passes based on which of our choices were available and which had the soonest window.
There is certainly some strategy/science involved in optimizing the FastPass system, but we found we could really make it work to our favor if we were willing to be flexible.
Hopefully, Disney will allow you to add your 4th pass via the MDE app in the future (if they don't already)... that would really save a lot of time/hassle...