I always add FO to my oils before the lye, mostly because my favorite recipe (co, pko, palm, shea. cocoa butter, olive, rbo, castor) moves quickly anyway; and I get in a hurry to get it in the mold and forget the fragrance.
Secondly, with SOME FINICKY FO'S in my experience it CAN mitigate seizing, ricing and soap on a stick as the FO is already diluted in the oils. One in particular, jasmine; will actually form a "tunnel" of solid soap followed by soap on a stick in the pot if I pour it in at emulsion. No colors or swirls for that one.
I HATE to make that soap, but I do it for a friend who loves the fragrance. With really badly behaved fragrances like the afore mentioned jasmine the mitigation of FO PITA is minimal, but its enough to at least get it in the mold.
I have noticed with some FO's that it SEEMS like the lye eats up some of the fragrance, or morphs it a bit; but those have been few and far between. Awupuhi Seaberry I'm looking at you here.
On the other hand, some FO's can make it harder to get to trace or even reverse what appears to be trace when added this way. For the record, I don't stick blend. I always seem to over stick blend so I switched to whisking by hand and get better results that way. Satsuma comes to mind as an FO that slows down or reverses trace for me; but it seems to do that regardless of when I add it.
As always ymmv.