I had previously seen the name “Caledonian” adjacent to the Chapeaux Cloche I find to be a most sensible style. In this case, the hat was more petite with an abruptly turned up brim. I found it to be closer to a toque style hat.
The black brimmed and white crowned Caledonian from the August 1861 Peterson’s Magazine essentially takes the fashionable shaped hat of the era and turns up the brim edge. The brim retains its curve while turning up substantially all the way around.
This type of turned up brim appears in fashion illustrations with other names attached as well. In Madam Demorest’s publications, we see it called the Burnside Riding Hat and the Boulevard.
This turned up brim does appear in photographs as well. I was pleased to acquire an image with a nice example of this upturned brim. This image shows how the brim is turned up significantly while remaining shaped all around. The brim dips in a curve in the front, and dips in the back.
This is a hat I need to replicate. I am thinking through how to develop the upturned brim while fashionably shaping the brim without a brim block in hand. I anticipate some wiring hidden further in the brim, as well as on the edge.
If I Had My Own Blue Box: 
Interesting picture. Are those rosettes on the ties?
Basically, yes. They show up on occasion in illustrations and photos.