This is a sequel of the work done on (strongly) monotonically monolithic spaces and their generalizations. We introduce the
notion of monotonically κ-monolithic space for any infinite cardinal κ and present the relevant results. We show, among other things, that any σ-product of monotonically κ-monolithic spaces is monotonically κ-monolithic for any infinite cardinal κ; besides, it is consistent that any strongly monotonically ω-monolithic space with caliber ω1 is second countable. We also study (strong) monotone κ-monolithicity in linearly ordered spaces and subspaces of ordinals.