I have come to the conclusion that the scarf is not going to be wide enough; it’s just over 4 inches and that’s not going to do it. I can ( and probably will) block it out a little - and that will help - but it needs more than that.
In experimental fashion, I knitted a swatch of stockinette with slip stitch borders (which is what I typically do on scarves). When it reached a reasonable size, I picked up the other three sides so I could experiment with knitting a border.
The first issue was how many stitches to pick up. I figured three out of four - or three quarters of the number of rows. I had been hoping that I could simply pick up the slipped stitches along the side but that wouldn’t have given me enough. I think the slipped stitches only came out to about 2/3’s. Anyway, I got that figured out and got the right number of stitches picked up.
Then I went around the circumference of the swatch piece for several rows, increasing 2 stitches in each corner, every other row. This unlovely but instructive piece is what I came up with:

Funny looking swatch
Of course I realized afterwards that making the center in lace or just some kind of patterned stitch would have helped distinguish the middle from the edging but so be it.
The corners look lousy because I was playing with yarn overs and various other methods of increase. However, the basic idea - 8 stitches every other round seems sound. The edging is straight with no pulling or puckers. What is bothersome is the picked up edges. The right one isn’t too bad but the left one is atrocious and sloppy looking. Is this a result of the slipped stitch edge, I wonder? And if I try this on the scarf, am I going to get something that looks this bad?

Scarf edge
I think part of the problem is that the swatch is in kitchen-type cotton which can be sloppy looking because it’s so inelastic. The edge of the scarf looks nice and even and not loose at all. I suppose the only way to know will be to try it. My other thought would be to crochet an edging (perhaps a ruffle?) if picking up the stitches to knit proves disastrous.
Oh, and I can now make a formal introduction, thanks to all your thoughtful and (sometimes) hilarious suggestions. I am pleased to present:

Agnes Ewe-lalie Woolley
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You are so disciplined with your swatching. I wish I had your patience (and your skill).
Pam - I really hasn’t anything to do with discipline. Anyone who knows me will tell you I am one of the most undisciplined people around. It’s just that when I want to find something out, I don’t want to experiment on a work in progress (lest I seriously screw something up) nor do I want to risk losing a substantial bit of good yarn. So a separate swatch is required :)
It was a mistake to try this with the cotton though, as it doesn’t behave in any way like the wool I’m using for the scarf.
