I've used sodium lactate, salt and bees wax for a harder bar. Then I found out that what I really wanted was a bar that lasted longer, not just harder (as in faster out of the mold and can cut earlier). All three cut the lather to some degree.
When I found out what I wanted was longevity, I got rid of most of my coconut oil and started adding rice of some sort to my soaps. I'm down to one bar of
@Dawni 's triple rice soap so will be doing another couple of batches this month. It makes a lot of difference in the longevity of my soaps. This time I will be making ice cubes from the rice water and rice puree, so I have that part made...sort of like master batching my lye water. And I love the feel of the lather on my skin better than I did before going to rice.
I finally bought some sorbitol and will be trying that to increase the bubbles, but since I already like the feel of the lather, the sorbitol is only for my grandson who loves bubbles, not lather.