I recently donated some baby blankets to my local hospital. While I was there, I asked if they need any other knitted items.
They said, "Yes, we could use some baby hats....NOT the tiny ones, though." They said they have LOTS of tiny hats, and they rarely get used. The bigger hats for normal sized newborns are what they really needed.
So, one of the nurses gave me a hat to use as a sample.
Here's the pattern I came up with from that hat:
(click the title to go to the pattern)
Stretchy Ribbed Newborn Hat - In the Round