Making this as a scarf that can button into a cowl, one that can be wrapped twice around the neck.

Made hat to match. My own, I referenced the hat pattern listed above for size, and changed the band, modifying the stripes with some texture to match the cowl pattern.

Hat brim folded back (wrong) in smile picture, but I love how happy he is with the surprise gift!

The hat turned out exactly the way I wanted it to! Not bad for a guess (size) and some new design stuff I wanted to try out.

MATCHING HAT:
Size 8 needles

In the round:
Red tuck decorative edging:
Provisional cast-on 80 sts (or other even number) with CC. K four rows (stockinette), fold, and K provisional stitches with current row, K2tog to end. (Join yarn here for knitting in round, or join after tuck is completed, when you begin MC, and sew the tuck ends together as finishing.)

Thermal ribbing brim (very warm!):
Join MC
Work 3 inches in Double Stockinette:
Row 1: K1, sl 1 pwise
Row 2: p1, K1
Each slipped stitch becomes a K stitch in the ribbing row. Each stitch that was knitted in Row 1, becomes a purl in Row 2 (so the purl columns look like seed stitch)

After ribbed band is finished (in the round):
Round 1: MC, K one row.
Round 2: CC, K one row.
Round 3: CC, p one row
Round 4 & 5 : MC, K (2 rows)
Round 6 & 7: CC, repeat rounds 2 & 3
Rounds 8 and beyond: MC Stockinette

Worked flat:
Provisional cast-on 81 (or other odd number) with CC. Work four rows (stockinette), fold, and K live provisional stitches with current row, one from front needle, one from provisional, K2tog to end. (81 live stitches)

Work 3 inches in Double Stockinette:
Join MC
Row 1: K1, sl 1 pwise wyib, end with K1
Row 2: K1, p1, k1
Each slipped stitch becomes a K stitch in the ribbing row. Each stitch that was knitted in Row 1, becomes a purl in Row 2

After ribbed band is finished (worked flat):
Round 1 (RS): MC, K one row
Round 2 (WS): MC, p one row
Round 3 (RS): CC, K one row
Round 4 (WS): CC, K one row
Round 5 (RS): MC, K one row
Round 6 (WS): MC, p one row
Rounds 7 (RS) & 8 (WS): CC, repeat rounds 3 & 4
Rows 9 and beyond: With MC, Stockinette

Crown Decreases (for both):
K 8, k2tog (72/73 stitches)
K
K7, k2tog (64/65 stitches)
K
K6, k2tog (56/57 stitches)
K
K5, k2tog (48/49 stitches)
K
K4, k2tog (40/41 stitches)
K
K3, k2tog (32/33 stitches)
K
K2, k2tog (24/25 stitches)
K
K1, k2tog (16/17 stitches)
K
k2tog all stitches (8/9 stitches)

Break off yarn with tail, thread tail through tapestry needle, and run yarn through all stitches in the direction you would knit. Draw yarn snug to close hole, weave in end.

If these seem too complex, try this:
https://www.ravelry.com/patterns/library/astronomer

It’s similar to this, I just changed the band and modified the stripes with some texture to match the cowl pattern. I also started off with a CC tuck, as a decorative edge.