I use a BB bag for my DIY bridge. The bag is silnylon, sits at the apex of the suspension triangle. When I pack up I
0. disconnect the spreader bars
1. disconnect from the tree at the foot end of the hammock. If your line or webbing to the tree from the apex ring does not disconnect, then roll it up.
2. Starting at the foot end I fold/roll the hammock body so that it is about 10-12 inches wide.
3. I roll the hammock body and fold it in, walking towards the tree at the head end. I am making a sort of cylinder that is 10-12 inches high.
4. When I get to the end of the hammock body I wrap in the sides of the suspension triangle, and stuff the cylinder into the bag.
Something like that ought to work for the JRB bridge.