One cannot read Paul’s letters without being aware that the teachings he was passing on “by word of mouth” were the same as what he wrote about in his letters.
What did Paul teach “by word of mouth”? We need only read what he wrote. There was but one body of “teaching” and since Paul cannot speak by word of mouth, his teachings are passed to us by letter, Scripture.
The chief reason for attempts to create a second body of teachings, Tradition, apart from Scripture, is to validate unscriptural teachings and justify it by pretending an equal or superior authority to Scripture, a practice well illustrated by the Pharisees.
Paul was not simply teaching the same things by word of mouth that he wrote about in his letters. Read the Didache or other historical documents of the time. The rules around baptism, celebration of the Mass, the Church hierarchy were all discussed in detail in the first century. These details are discussed in Scripture only briefly but were very well developed outside of scripture.
Remember, Paul often focused on what was going wrong in the Christian communities he was writing. His letters did not serve as a manual for behavior - that was known by tradition and by things like the Didache.