Yes it is possible to pick up some FP+’s with short notice. Unless you happen to come upon one that has just been released you won’t find ones for FoP, Navi River, and 7DMT. A few of the “second tier favorites, like Peter Pan, Soarin’, and the Safari, are, in my experience, challenging but still possible if you can be flexible with the day and time of day.
DW and I are trying just this experiment for our parks visits this winter. Instead of getting up early for rope drop we are trying the FP+ route. I have had success booking about 7-14 days out. In one case, trying to get Peter Pan, I had to try a few times, looking at a couple of different days, before availability popped up in the early afternoon. After a quick confirmation I booked a couple more around that FP.
So we have traded getting up early, waiting at rope drop, then riding our favorites with low waits for busier parking lots, mid day crowds, and FP+ waits. DW appreciates not having to get early and then stand waiting for rope drop.
Over the years our visits to the parks have become more like walks with interesting interruptions along the way. So far the FP+ route has been working. I continue to hone my FP+ booking skills.
The only thing I’ll add is that you’ll probably note a few more people in the parks during what you might remember as “quieter times”.