I like to put tension rods in my windows and hang clothes on hangars from the tension rods. If you have curtain rods already, that would work. I find they get musty smelling indoors unless you hang them in sunlight. I have also been known to use the backs of chairs. You can dry socks and other small items by clothespinning them to a hangar.
My neighbours put chain link fence around their land, which is uninhabited, and we have found it makes a great clothes line. I like to pin things right to the side of it so they aren't folded, and dry really quickly.
Works at a residential alternative high school in the Himalayas SECMOL.org . "Back home" is Cape Cod, E Coast USA.
1. the quintessential 'Hills Hoist': generations of kids got into big trouble swinging on these. Most people have them if their backyard is big enough. Also, throwing a tarp over it means clothes can still be dried in wet weather. Some entrepreneur also manufactured a fitted tarp so it could be used as a sun-umbrella or shelter tables of food during parties!
2. the lean-to design: two sturdy posts placed in the ground as far apart as you like. A timber cross-member bolted to each post so it can tilt. Then string two or four lengths of wire between the cross-members and pull tight. Clothes are then pegged to the wires - a large stick or pole can be used to hold one side of the wire higher than the other e.g. for large items such as sheets and towels. (The photo shows a fixed cross-member version, which isn't as good.)
I would get whatever is left of your line and tie it to whatever you have around (maybe drive a post if you need to).
That lacking, when I was in college I washed my clothes in the sink and dried them on the fire escape on hangers. didn't win any awards for brilliance, to this day I've never owned a dryer and know that was probably not the easiest way to do it, but it worked just fine for 4 years, even in the winter.
Here people often hang their laundry on a wire fence or on whatever is stable (and clean) enough to permit.
I keep some old rope around to put up extra clotheslines when I wash bedding, for example. Any old rope or even wire will do just fine.
(I love the image of the tree drying the clothes, but I imagine your clothes might end up blown clean to Arizona....)