So, no more mystery - this is what my finished hat looks like:
The construction is really clever. You start by knitting a long stocking stitch tube to make the brim, and then join it together to make a circle. If you've got the row tension right (which I did), you then need to pick up one stitch for the body of the hat for every row of the brim. So you're picking up each strand of yarn in turn between two columns of stitches in the brim - that's very straightforward, and makes a very neat join.
The body of the hat is reverse stocking stitch with little circles made of cables all over. The cables give it a nice texture. Then on the crown, you decrease the stitches very rapidly, so that the top of the hat is gathered in to the central point.
And finally, you sew a button on to the brim, to hold down the tab where the ends of the brim overlap. I found a nice button at the Harrogate Knitting & Stitching Show.
Now that it is no longer a mystery hat, the pattern is called Encircle - you'll find it on Ravelry or here.
(By the way, the hat is not magically levitating - I put it onto a souffle dish, and then perched that on a jar. The combination made quite a good hatstand.)