one idea i just thought up just now, is to goto the material store and buy some thick white sheeting (thick so hopefully you won't see shadows coming thru on the other side or whatnot)
then head down to your camping store, or op shop maybe, or the sports store
.. and buy the cheapest TENT you can,
the rods that hold tents up are roughly a meter long, and they connect to each other like fishing rods OR better for this case - get the ones that are already connected with elastic, yet you can pull them apart and they will fold together yet not get tangled.
if the material you buy is a tad elastic, you can sew/hook/twine/ or better, put holes in your sheet at the top bottom and sides and thread the poles thru (in,out,in,out)..
that way, you have something in the trunk of your car that resembles a rolled up tent, and you can even use your tent bag to carry it
then on site, you can pull it out, give it a shake and a pull, roll it all out, and in 30 seconds flat it will be a tight uncreased backdrop that you can lay on the ground, or lift up and rest it against a tree or wall.. because of the elastic in the material and the elastic in the rods, it will go together with ease, and the sheet itself will be tort and stretched enough to be a dead straight backdrop!! 
note: when you thread the poles/rods through the material, think ahead and measure your material to be a little bit UNDER the size of your rectangle box (the rods all connected up). that way, the sheet will have to stretch when it's all put together..
in theory, it works. when i get the funds, i'll make one 
maybe i should patent this idea before i click "post quick reply".. meh.