Halliday & Matthiessen (2014: 417-8):
Figure 6-21 gives the analysis of a clause with a phrasal verb of the adverbial type (i) in it, in terms of (a) transitivity and (b) mood. Similarly with the prepositional type (ii): in I’m looking for a needle, the mood constituents are looking Predicator, for a needle Adjunct, and this accounts for the ordering relative to other Adjuncts, e.g. I’ve looked everywhere for a needle. The third type includes some where both adverb and preposition are (or may be) part of the Process, e.g. look out for, put up with, put in for; and others where only the adverb is within the Process, e.g. let in for, put up to, as in he let me in for it, he put me up to it.