Or ...
Auntie Clara did a bubble soap, which has the dipped soap idea that could be used for multiple layers ... it gives a really smooth, thin finish (and could be adapted for Melt and Pour, if you prefer to buy a pre-made soap base):
https://auntieclaras.com/2016/09/dip-n-drip-soap-beads-bubble-soap/
Or ...
Another possibility is to make cold process soap dough and use that to create the layers.
Make a single, uncoloured, batch of soap, then divide most of it into colour cups (where you have colour to mix into the soap batter in the cup already).
Let the soap set up until it's firm, but still really soft, and then either form them straight away (wear gloves) or put them into air tight bags.
The soap can stay in the bags for months and still be soft, so you can take out what you need and leave the rest in the bag - just remember to push out any air so the soap doesn't go hard.
Using gloves to protect your hands, make your first ball shape by rolling it between your palms ... let it harden a little, then take the next colour and spread it (roll it out first, or spread it directly on your ball) around the original ball and leave that to set up and harden a little ... keep repeating until you have all the layers you want. The layers will become one soap as it cures.
If you are worried about colour bleed, add an uncoloured layer between each coloured layer.