I expect I would have to put down many coats to do the job. (SOURCE)
One factor to distinguish phrasal verbs from prepositional verbs is particle movement. Phrasal verbs can place the particle before or after direct object, whereas the preposition in prepositional verbs must precede the noun. (Biber et al. 1999)
So, I'm confused when analyzing the fragment "put down many coats to" because I'm not able to distinguish whether "put ... " is a phrasal verb or a prepositional verb. The confusion arises when I read in the Oxford Dictionary of English, under 'put' → 'phrasal verb', the form "put something down to" is correct.
So, the above sentence would be rewritten as follows :
I expect I would have to put many coats down to do the job.
Am I right? If not, how do I correctly distinguish phrasal verbs from prepositional verbs in the cases, like this one, where there are two particles (in this case 'down' and 'to')?