I'm not sure what your stitch looks like or what a long loom is or does. But I will say that stockinette stitch curls. Even putting a border on it it can still curl some.
If your stitch looks like bumps on one side and smooth V's on the other that is stockinette so it will curl. Think most of the Dr. who scarves I've seen have been garter stitch which won't curl. Some may have been knit in the round.
Garter stitch knit flat is all knit stitches on both sides.
Stockinette knit flat is one row knit, one row purl. Knit in the round it's all knit.
So it depends on your loom. Does it knit flat or in the round.