Wow, I've been absolutely loving the process of tweaking my baby hat designs for summer. I'm sure lots of people might say "duh", but using those old standby Crochet Cotton Threads instead of yarn is WONDERFUL for a summer option!! Try working your pattern made for a baby (lightweight or no. 3 or "sport weight") yarn in the SAME size hook -- NOT a thread crochet hook. You'll have to use a double strand of the thread, but that's easy. Above you can see how my classic granny baby hat in crochet looks when you work in with a double strand of thread instead of baby yarn. It feels so heavy and supple and cool when worked in that double strand of no. 10!!! Love it! Heavier cotton threads like no. 3 don't work as well, they're rougher.
The only down side is a loss of springiness, so you may wish to add little ties that tie under a chin. That's why I'm working on that brimmed hat you may have seen in the previous post that incorporates a tie that works well with the hat. That less flexible thread hat might just scootch and scootch until it's off that darling little head, and THEN what good is it, I ask ya?
Here's the pattern you see above on Ravelry - Granny Hat in Newborn / 3 mo / 6 mo ;