When I first saw the photo of a group of women posing with the croquet game and men, I squeeled in glee at the hats. I love these hats, both the shape and decoration. The cool thing about this dome shape is how it has great balance when you wear it.
You can find this hat, with its grape purple velvet ribbon points, in my shop.
https://www.etsy.com/listing/685183020/civil-war-era-straw-hat-hand-sewn-by
You may have noticed I am posting more millinery than usual. I am on “spring break”, which gives me more time to sew and also happens to be mostly unpaid requiring me to sew more. Good thing I enjoy this. Originally, my goal was to make a few extra pieces. Then, the idea of being able to offer a new piece each day of break began to appeal. We shall see how it goes.
If I Had My Own Blue Box: 
Absolutely gorgeous hat! Very sporty but could it be used as a summer hat too?
If you mean for the 1860s, this shape does show up for other occasions. You would be safe wearing it for time in the garden, a country visit, and similar recreational activities. If you mean in general, I know one friend wears her’s of the same shape for modern wear as well.