When I made the Petal Baby Hat, I followed the pattern directions for a preemie hat and cast on 63 stitches. I used Caron Simply Soft, with size 6 and size 8 needles per the directions in the pattern. For me that ended up being a hat for a baby's head circumference of about 13 - 13.5 inches. When I left out a repeat to make a smaller version, that changed the height of the hat, but not the circumference. I had still cast on 63 stitches. It just seemed to me that my first version would be able to cover up the newborn baby's eyes (and maybe even nose)! But I'll take them both to the hospital in a few weeks and find out.
Of course all babies will be different, but one of the knitting books I have lists these guidelines for baby hat measurements & sizes:
Age 0-3 months: 13.5 inch head circumference
Age 3-6 months: 15 inch head circumference
Age 6-9 months: 16.5 inch head circumference
I'm guessing that if you followed the regular pattern directions for a newborn hat and cast on the 73 stitches (instead of 63), it will be close to what you want for a 3-6 month old. This is a super easy pattern, and I was able to finish it very quickly, so it's not a big time investment to play around and figure out how to get the hat to be the size you need.
Good luck! I'd love to hear how yours turns out!