I'm finding that timing may be much more important than I had thought--the results have been quite varied. The LSU Improved Celeste and LSU Scott's Black grafts I did three months ago (March 10) are doing great. The LSU IC added 14 inches and is still going strong, with several figs forming. On the other hand, the Black Madeira put out a few wimpy leaves and hasn't done anything since.
The grafts I did on May 4th, after it was already hot, haven't progressed at all. I don't know if this is due to the advanced state of foliage on the other branches, or the temperature, or some other factor I haven't thought of. Maybe it was just beginner's luck on the successful ones--I hope not! (At any rate, 100% of the persimmon grafts "took" and have grown like weeds.)
I'm still guardedly optimistic about the grafts that don't seem to be doing anything, because another graft (Violette de Bordeaux) that I thought was dead is just starting to shoot out a leaf--but I'm guessing not much will happen with the others till the summer monsoon's typical growth flush, or maybe not even till next spring!
So, my best guess, based on very limited experience, is that fig grafts should be made just as the recipient tree is breaking bud, or very soon thereafter.