The earliest mention of the chapeau à la Spa explains that the hat was created in Spa and moved from there to the French court. It was worn by the Duchess of Devonshire, a repeated visitor, and was also called the chapeau à la Devonshire. In 1779, though married, she stayed at Spa with her mother and younger sister; the Duke was preparing soldiers for the war against America and France, but at Spa aristocracy from all countries could mingle on neutral ground. Georgiana was specifically noted as having a "particular admirer" in Mme de Polignac, who is presumably the one who brought the style to court. (See: Amanda Foreman, Georgiana, Duchess of Devonshire.)
The Spa hat was of black straw, cocked three times. Bows decorated the cocked sides, and a ribbon ran around the crown. Ostrich plumes in varied colors were a must, but the flowers were optional and possibly not usual.
It's notable that the chapeau à la Spa only seems to appear in fashion plates with "oriental" styles - collared Levites, polonaise jackets. These are of interest to costumers, but don't appear frequently in collections, because they were fairly short-lived trends and had the strong class connotations dressing very fashionably carries. Appearing in fashion plates with these marks the Spa hat, despite its relatively simple form, as being an elite trend.
It was also a short-lived trend. I can only find it in plates from 1779 and 1780, and no references in other fashion periodicals. It would certainly be an interesting style to replicate, however!
Edit: While I can only find the actual chapeau à la Spa in those years, there are other fashion plates and paintings that show what appear to be later variations, very similar except that the hat brim is cocked only twice, or not at all.