Thanks -- your updates are helpful to me.
The added fluidity of the hot soap was probably because you cooked enough water out of the soap to slightly change the properties of the gel. Soap in the gel stage can be more or less viscous depending on water content, fatty acid content, and temperature. It may seem counter-intuitive, but if soap is in full gel, it can be more fluid if it contains somewhat less water, all other things being equal.
People who make soap with a boiled method (or those of us who use the salting-out method of cleaning up soap) can observe this viscosity change, but I don't think people who make HP or CP soap have many opportunities to learn how the properties of gelled soap change with water content (again, all other things being equal.)
You can add fluidity if you make HP soap starting with less water and being scrupulous to prevent water evaporation around the edges of the soap while cooking it (the big downside to using less water). You don't need to cook for hours. The SMF "sous vide" soap challenge a few months back might be worth a try if you want to make a lower-water HP soap without cooking it to death or getting lots of crusty bits.
Also a finished room-temperature soap with a lower water content feels harder and denser than the same soap with more water. Do you only use a hot process method to make soap? If so, you're probably used to having a fair bit of water in your finished soap, which makes it soft and spongy. You might want to try some cold process batches with higher lye concentration (33% to 40%) to see the difference in texture that less water makes. Or make some sous vide HP batches using the same 33% to 40% lye concentration.