The dough is supposed to be heavy and wet. Here's the original article which refers to the wetness of the dough.
"He mixes a very wet dough, about 42 percent water, which is at the extreme high end of the range that professional bakers use to create crisp crust and large, well-structured crumb, both of which are evident in this loaf. "
"By starting this very wet dough in a hot, covered pot, Mr. Lahey lets the crust develop in a moist, enclosed environment. "
It really shouldn't need much more flour to get it to the proper consistency for the second rise. It is not supposed to be "like a baby's bottom," which is the descriptive standard for a kneaded bread. It's supposed to be a wet dough which has enough extra flour added to hold its shape.
I've made this bread at least 2 dozen times at this point and have had to add a significant amount of extra flour a few times to get the dough to hoild shape, so I did some subsequent research on it. IMO if you let the first rise go much beyond 16 hours you run the risk of having some of the water "weep" out of the dough. I don't have my Cookwise right here but it says that if the dough is left to rise for too long, some of the water can be forced out of it.
I have experienced that myself with the dough going from shaggy and heavy and bubbly, etc at 12-14 hours to somewhat separated at 20 hours. With a 24 hour rise there definitely was some water that had separated from the dough. Under these circumstances it was impossible to get the dough to keep it's shape without adding maybe 1/3 - 1/2 cup more flour.
So now I try to let the first rise go 16 hours and no more.
Also -- skip the towel and use a bowl like others have said.