The leniency for leaving up to a tefach of hair uncovered is based on a teshuvah by R Moshe Feinstein and has wide halachic support. Covering only rov rosh (most of the head) is widely understood as satisfying the Torah-level obligation of hair covering, but not the rabbinic level.

For sources, please see the relevant chapter of R Getsel Ellinson's The Modest Way. For a more developed discussion of what may or may not be left uncovered, please see R Yehuda Henkin's Understanding Tzniut, pp. 33-42, in particular his treatment of Igrot Moshe.