I know that solar stills aren't worth much in the dry sand dune desert, but they can be very useful for turning saltwater (beaches, saltmarshes) into fresh, or for getting water from vegetation (even wrapped around limbs of trees and bushes, trapping transpiration), or from non-desert areas with no other water source other than a slightly moist soil, and even to get clean water from a contaminated source.
But even in the dry desert, they are better than nothing. They also can be used to turn urine into fresh water.
The thing with solar stills in the desert is to fill them with vegetation (cactus or whatever is available) and urine.
Despite the negative press a solar still can have, they actually have many applicable uses. We just typically tend to think about the desert when we hear someone say solar still.

And also, the sheet can catch alot of rainwater.

I might have to give that sponge idea a go. Kept in a plastic bag (to avoid unwanted moisture gain) in the inside sleeve might be just right. Although the bandana could probably serve that purpose just as well. Maybe I'll do an experiment comparing the two.