I was intrigued that the instructions call for very narrow elastic to be threaded through the wrong side of the top and bottom edges.
I like to use elastic in some of my hats and headbands. It's a way of ensuring that the hat/headband edge stays the same size -- especially if the yarn is not very stretchy (such as cotton, silk or rayon) or if it has poor recovery or tends to grow (such as alpaca).
By the way, the very tall stitches in this vintage headband pattern are treble stitches with extra loops on the hook. You start this stitch by wrapping the yarn over the hook 4 times (instead of the usual 2).