Stephen,
I had same issue with my Chicago Hardy, full of green figs, putting on some very tall branches so Today I tried to put on air layers from those three very tall branches with green figs. Figured if lucky and can remove them in a month it still gives me some time to pot them up and give to friends. I know a bit late in the season but might be ok. I trim my CH around November before I wrap it. CH is an amazing tree and requires so little care and rewards us with hundreds of sweet figs. Even when it dies back in tough Long Island winters it always roars back. Good luck.