We consider Kazhdan-Lusztig cells of the symmetric group Sn containing the longest element of a standard parabolic subgroup of Sn. Extending some of the ideas in [Beiträge zur Algebra und Geometrie, 59 (2018), no. 3, 523–547] and [Journal of Algebra and Its Applications, 20 (2021), no. 10, 2150181], we determine the rim of some additional families of cells and also of certain induced unions of cells. These rims provide minimal determining sets for certain W-graph ideals introduced in [Journal of Algebra, 361 (2012), 188–212].