Gosh, I haven't yet run into this problem, yet. I have in RL until we could afford our own dryer. What we had to do on rainy days was to use an indoor drying rack. Happily, I had a big enough kitchen for it.
As for what you might try there is a glass roofed awning. Or, perhaps you can created a building out in the back yard with a high enough ceiling? Like a green house, without plants.
This sounds like a situation from the 18th century. Sometimes, a family had to wait for the first warm day after winter to do any laundry at all. Clothes were hung over bushes to dry! If you had a baby in the house, their napkins were dried in front of the fireplace. Can you imagine the stench?
