How to store bread dough in the fridge for proofing?
Using the refrigerator for the first rise is absolutely easy, but that is not what you are asking about.

For a second rise, tweaking your current approach a bit, I suggest:


Place yout bread(s) on a baking tray or similar or use a well-floured banneton (basket).
Find a plastic bag large enough to fit the entire tray or the banneton plus a bit extra room inside.
Lightly "blow up" the filled bag and close with a clip, rubber band or similar. You are not creating a "ballon", just enough to have the bag "float" a bit over the loaf.
Another option would be placing a glass, can or similar object next to your loaves to support the foil tent from below.
You need not create an air-tight seal, but you want to limit the loss of moisture to keep the surface of the bread from drying out. 
For a single boule, an inverted bowl can protect the loaf enough, or if you use an appropriately sized Tupperware or similar bowl in lieu of a banneton, just  close the lid.


A few additional thoughts: 


For round loaves (boules) a banneton or in a pinch, a bowl, helps to keep the loaf in shape. Especially for long proofing times the tension created by shaping the loaf might not be enough to hold over hours, causing the loaf to "flow" sideways. Flour the basket well and don't worry about excess flour at the bottom (or top, depending in technique) of the loaf, simply brush it off after baking. 
You don't need to put parchment under the loaf if you transfer them to a pizza stone or pre-heated tray. Don't try to lift a baguette by hand, use a sword-shaped baguette peel or, in a pinch, a cutting board and roll the baguette onto the peel and back onto the pizza stone or second tray.
That said, damp paper isn't much of a problem unless it is so wet that it will simply tear.
Rolls or other small items are easier to pack into a plastic container - plus they bake quickly in the morning, which may add a bit of convenience (or allows for a few extra minutes of sleep).