You can try, next time, using the clear plastic sheeting they sell to cover tables and chair seats with at the dollar store--or even painter dropcloths in clear pastic. Layer it over the fabric, cut to fit and then sew using a metal foot on the machine using a zig zag or similar stitch on the edges. I did this in the past with nylon sewing thread, to make dining room chair seat covers, so my young son couldn't ruin them. If the plastic is thin enough it sews easily but you definitley need a metal foot to hold the layers together as you sew--the lightweight plastic feet just won't do.
I like your idea of using clear contact paper, as it is thin enough and should stick to the fabric w/no slipping as you sew.
Let us know how the project went!