When I knit the Harry Potter scarf for my son a few years ago - I knit it in the round.
Stockinette likes to curl, usually the way that is controlled is by knitting a round or two of rib, garter, purl. To keep it from curling in on the sides - moss/seed or garter a few stitches on each side of the scarf.
If you go back to knitting in the round, use DPN's and cast on fewer stitches to reduce the bulk. You may have to play around with it a bit till you get the desired effect.