The lufa idea is clever, but a bandana has 1000 uses. You might be able to bundle your tarp with a bandana.
I use a “bishop bag” for hauling my hammock and tarp. It is a mesh drawstring stuff sack with one of the bottom holes widened a bit. That slips over the head end of my whoopie sling suspension and I can stuff hammock and tarp in it. It is light and has good ventilation to aid drying. I put my tree straps in last so deployment is to put up the head end tree strap, clip in the whoopie sling and pull out the hammock and tarp as I walk to the other tree, hook up the foot end and then start hanging the tarp. The bishop bah just sits at the head end. It aids stalling drips too. If it is raining I can still get the tarp out and hang that first to keep me and the hammock dry. I like having it all in one bag— forgetting your tree straps or the like sucks :)