  We consider Kazhdan-Lusztig cells of the symmetric group $S_n$ containing the
longest element of a standard parabolic subgroup of $S_n$. Extending some of
the ideas in [Beitr{\"a}ge 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].
