I wouldn't use it for a backing, simply because it wouldn't be seen. Most tartans are wool...and they're incredibly expensive! I know that many years ago, my mom bought one yard of her beloved MacKenzie tartan...for $23.00 a yard! *flinch* I agree with Garnet...the scarf is a great idea!
I'm sad that I have no idea what happened to the piece of fabric...I know she never used it...