My advice would be to do it now or in early spring.

March.

Other than the heat and dryness of summer, there is really not a bad time to transplant these. They are extremely adaptable, resilient and very tolerant of transplanting, provided you are moving them to a situation/location they appreciate. Personal experience and even the advice of local native plant guru Art Kruckeberg would indicate that any time aside from the dry heat of high summer is suitable.

I have moved likely hundreds and fall transplanting has always gone better. The small ones are no big deal whenever.
But if you are talking about a really big one--DO IT NOW, NOT IN THE SPRING.
They have reestablished MUCH better when moved in the fall than in the spring. The issue is keeping them watered during the first summer. But of course it depends on the conditions where you are relocating them.

My experience has been to try to get them growing in fairly dry shade and spring transplanting has never gone well in those conditions.

That would be consistent with other hardy plants, of course. The convention that end of winter is best could certainly be wrong. Spring as in when fronds are unfolding would certainly be undesirable.

I have moved them into dry shade under cedars and had to water every week in the summer, but I moved some to near the bottom of a steep bank with a ditch at the bottom that has water in winter, and didn't water them at all afterward. I generally transplant both in fall and spring, I don't remember when I moved sword ferns.