If outside, and it is practical...I would bring them indoors and not worry. Keep damp for the first month (you should have buds and even small leafs growing by then) and then water only after the soil seems to be drying. Growth indoors is good and should provide you with a good root system to plant during the coming Spring. Figs don't have to have chill hours as most fruits require.
If outside and are going to be left there, I would cover well to prevent freezing. Water sparingly, you do not want the cutting to grow new growth outside during this period.
The picture here are cuttings I took from a broken limb the first week of September. Already budding out in a five gallon bucket.

Thank you for that valuable info!