I 'had' to buy thin fabric once to get the shade of tan I wanted. Fusible interfacing made it work well.

I agree with the suggestion to fuse it. I like Misty Fuse as it is less stiff than other fusible webs. I've fused old, thinning fabric to inexpensive muslin successfully. It can sometimes get tricky if the lightweight fabric is not 100% cotton (speaking from experience, Pellon 805 is not designed to fuse anything other than 100% cotton and it was a brutal, long process trying over & over again before I finally got the fabric fused properly), but as long as you are sure it's all cotton, fusing is a wonderful option.