This is a continuation of the author's earlier paper with a similar title [\textit{C. Xi}, J. Pure Appl. Algebra 193, 287--305 (2004); erratum ibid. 202, 325--328 (2005; Zbl 1067.16016)]. The setup consists of an algebra monomorphism \(f : B \to A\) or a chain of such monomorphisms. In the case of a chain the author assumes that, at each step, the radical of the subalgebra is a (left) ideal of the ambient algebra, which is assumed to be of finite projective dimension as a module over the subalgebra. In the case \(A\) is of finite global dimension the author shows that the finitistic dimension of \(B\) is finite. Another context considered by the author imposes an additional condition on the radical of \(B\) that its extension (as a right ideal) along \(f\) coincides with the radical of \(A\). In that case the finitistic dimension of \(B\) is finite, provided the global dimension of \(A\) is at most four. Another result of the author gives, under the same assumption on \(f\), the finiteness of the finitistic dimension of \(B\) provided the representation dimension of \(A\) is at most three.