Factorizations of natural embeddings of l_p^n int L_r

This is a continuation of the paper [FJS] with a similar title. Several results from there are strengthened, in particular: 1. If T is a "natural" embedding of l_2^n into L_1 then, for any well-bounded factorization of T through an L_1 space in the form T=uv with v of norm one, u well-preserves a copy of l_1^k with k exponential in n. 2. Any norm one operator from a C(K) space which well-preserves a copy of l_2^n also well-preserves a copy of l_{\infty}^k with k exponential in n. As an application of these and other results we show the existence, for any n, of an n-dimensional space which well-embeds into a space with an unconditional basis only if the latter contains a copy of l_{\infty}^k with k exponential in n.