It is often thought that an awareness of the scripture knowledge of the specific audiences to which Paul writes (including 2nd Temple Jewish literature and interpretive traditions) will assist us in establishing this, but I have some doubts. The general line of thought is as follows: if the recipients of a particular letter are unlikely to detect and/or understand a particular "allusion" to scripture within the letter because they are not sufficiently familiar with the scripture alluded to, then it is unlikely that Paul intended such an "allusion".
Apart from the difficulty in establishing the general amount and nature of scripture knowledge and the amount and nature of scriptural knowledge in specific churches, this line of thought rests on the twin premises that Paul is sensitive to the scripture knowledge of his audiences and so places constraints on his references to them (including "allusions") in his writing. As far as I know these are assumptions lacking explicit evidence in his writings (Could 1 Peter 3.15-16 refer at least in part to scriptural allusions?). Furthermore, what amount of scripture knowledge did Paul feel constrained to, and how would we determine this? Did he know, assume, or expect that there would be those present at readings of his letters with adequate scripture knowledge (perhaps the 'informed' or 'competent audience' in Chris Stanley's terminology) who could point out or explain references to the scriptures?