I have a main crop that has formed and are about 1/2 their expected size right now and quit green (on my unknown fig tree).

There is additional growth on the branches past where the fruit has set. Should one remove all the recent branch/leaf growth of the branch past where the fruit has already set this late in the season?

The new growth looks as if it may produce a few additional figs on the new growth but I have never seen those guys out on the ends ripen up before the cold sets in.

What do you folks do? Last year I cut 1/2 them off and it seemed to help the remaining fruit ripen up.

I always cut the ends of fruit bearing limbs after about 5 sets of leaves give or take a few sets depending on growth habit on plant in my climate.
When i say 5 sets i count the pair of leaves as "1 set" as they grow on opposite sides of branch.