I poured about a dozen tall ice-cream buckets of water on the pool area. This only managed to wet down about half of the depression left by the pool. This will take a lot more water than I had thought. It wet through the snow and slushed up on the mud. I am hoping that it will freeze up overnight and allow my second and subsequent coats to fill in the area a bit more evenly.
If this doesn't work as I thought, I will stop without doing the entire area. I can now see why they usually flood the rinks with a hose. If this does works, maybe next year I will bring a hose inside to thaw the ice and use that to flood the rink.
That is, as long as the use of the rink outweighs the annoyance in making the rink.