Sorry it took me a while to get these up, hope they are helpful. This is the baby hat my daughter got from the hospital when she was born. Although I think it was a bit small for her, it was always popping off her head! I think it would have needed another couple centimeters/an inch in length (actually width as you were knitting) and another set of 4 rows of stockinette/reverse stockinette to fit her better (8 altogether).
It's approximately 20 cm/8 inches unfolded from bottom edge to top, which would be the width as you knit. It it alternating four rows of stockinette and reverse stockinette x 30. Or maybe I should phrase that as 4 rows stockinette then 4 rows reverse stockinette x 15. I think it's a fingering weight yarn, possibly a baby sport weight. I tried to get a picture of the seam and how they gathered up the one end on the inside, but it's not too clear. I can tell they slipped the side stitches that eventually became the bottom of the hat.