With one of the salvaged shell sections selected,  I worked on a number of overlapping tasks in repetitive iterations - looking at orientation, seat height, seat angle, sketching some reference lines, etc.   The piece is not symmetric about any plane, but I think any asymmetries won't be too noticeable.  And if they are, I'll say that it's part of the charm.

A milk crate worked well as a staging platform for testing height and angle.   Some scrap wood helped to stabilize it enough to let me do some test sits to find the most comfortable position.  This is obviously a corner section from the hot tub, but it's been inverted - what was the seat is now the back.

Once I was satisfied with the height and angle, I made a number of measurements of dimensions and angles that I could use to sketch out a base.   

The prototype would be built from scrap plywood as a proof-of-concept, and if it worked, I'll replicate it in white oak.  To contrast the flowing curves of the shell, I went with straight lines and hard angles for the base.  The design had planar side assemblies that would be connected with front-to-back stretchers that would shaped to support the shell.

Checking the angles after a domino dry fit.

After that, I spent a lot of time playing around with different stretcher lengths to arrive at this.  The rear stretcher is shorter than the front and the leg assemblies are canted-in at the top.  This made for some frustration on cutting the compound angles on the ends of the stretchers.

Before glue-up, I shaped the excess along the top edge of the side rails to create pillars that would support the shell.

These received some further shaping on the top to follow the contours of the shell.

With the base ready for glue-up, I turned my attention to marking and trimming the shell to final shape.   A section of electrical fish rod was perfect for getting a smooth curve in three dimensions.

Rough-cut to shape.

Edges sanded smooth and mounted to the base with some screws.

The seat was comfortable and the thin plywood legs held without complaint.  The seat was just a little high, but that would be an easy adjustment.   In the end, though, I wasn't crazy about the look of the base, neither front nor side.  Maybe it was too spindly looking from the front?  And the hard angles on the sides seemed too jarring.

I did start down the path of an second prototype that had planar front and back assemblies.  Since the face of the wood faced forward on the legs, I thought this would address the spindly look.  I'm not sure it did.  It made me think of a bow saw, and once I had that image in my head, I couldn't see it as anything else.   The compound miters on the front-to-back connecting rails were giving me fits, and I realized this design wasn't all that much of a visual departure from the first, so I abandoned this approach before completing it.

At this point, I set aside the project for a bit to mull over some options and wait for inspiration to hit.