The little clothesline falcon suggests is brilliant!
We used ours when the lines were full.
You can also make one quite easily if you go to the fabric store and purchase round elastic (1/8"), maybe about 20 feet. Then double it, and tie the ends together. Put one end over a chair top or doorknob, pull it tight but not stretched, and you twist the other end until it is so tight it begins to double up on itself. At that point, you carefully have a 2d person hold the middle, fold it in the middle and take your end to the doorknob ... tie the ends together... and let it go!
You can also try just twisting and doubling it once if you can only find 1/4" elastic rope.
We've made these many times and they work great! You can put caribiners or other clips on the ends. You don't need clothespins with these. You just stick the corners of your clothes up into the elastic and it holds it tightly.
You also can take some baby diaper pins or large safety pins and pin your clothes to your backpack in the morning if they are not dry. They'll dry as you walk.
There were only 2 or 3 times we didn't find a clothesline, though.
Another tip: Pick up a bar of Naptha soap when you get to Spain and cut it into 3 or 4 pieces. Share with other pilgrims. It works best for handwashing your clothes - it's made for that.