I want to start off by making it clear that this is not my pattern. The info I have on it says it's from "Ann in Tenn" and has an email (though I haven't tried to contact her). The printout is dated January 26 2006 and was a message in a Yahoo group, most likely the ISM Club. I'm reprinting it, along with the gauge and size info that I got on my machines.
Worsted Weight Yarn
ISM/USM: KP 3 with 8 wedges on 70-72sts or KP 4 with 7 wedges and 58 sts
SK155: Plymouth Encore, 54gr, T5, 58st, 7 wedges; 18sts to 4"/23 rows; resulted in a toddler size aprox 16 1/2" circumference
LK150: LionBrand Fisherman Wool, T5.5, 70st, 8 sections; 20st to 4"/23 rows; don't know the exact size but it was too small for a 20 1/4" head.
KP 2 with 7 wedges over 68 sts and short row the end 16 sts (instead of 12).
Use an open cast on, and cast on 72 sts. One layer will get folded into the other, resulting in a hat that's 36 sts tall. 12 of these are for the crown and 24 are for the length down to the forehead. If you want to adjust the length or add a pattern, do it in the 24st section on both sides of 0. The hat that I made would have only a tiny brim, but I do not have a toddler to try it on.
Knit one row.
Put a needle into hold at each end of the bed, knit across. Repeat this row until there are 12 needles in hold at each end and you have knitted the row after pulling out the 12th needles. Make sure to use claw weights.
Place the needles on the far end of the bed into work position and knit across. Repeat.
This makes 14 rows for the section, plus the original first complete row.
Knit 7 or 8 wedges the same way (not including that very first plain row).
Take it off on waste yarn, and graft. Use the yarn ends to close up the little holes at either end.
There will be one row for the grafting, so the total number of rows will be 7 (or 8) x 14 + 2.
You may, if comfortable with grafting, eliminate the first plain row and the very last return row on the last wedge.
You can use an intarsia technique to make the inside and outside different colours. Change colours between L1 and R1. You can also change colours for each wedge. Or do intarsia and make the center 16 sts a different colour for a contrast brim.
In the gauge I got, adding an 8th wedge would add 2 1/2" to bring it up to 19", still a small child size; a 9th wedge would make it 21 1/2", an exact fit for my head which is not really a good idea; you want a little bit of negative ease. I think I'd try T4 for a wedge and see what that tension gives. Nine wedges of 14 rows makes a total of 126 rows; I could also try doing 7 wedges with 18 rows. To keep it to 12 sts at either end, instead of one stitch to hold at each end on each row, I'd have to do a little figuring. I do think that the sections are a little narrow which results in a pointy hat.
I'd also add about 18 sts in the middle so that there would be a fold back brim.