What makes you think that Paul preached only to believers?....more than not, I am sure you would agree that the majority of his audience were those who did not believe.
Who did Paul write the epistles to? The church. The saints. As far as I know, Romans is the only
epistle that could be considered partially for unbelievers. How does Paul start off his letters?
1 Cor 1:1; Paul, called to be an apostle of Jesus Christ through the will of God, and Sosthenes our brother,
1 Cor 1:2; To the church of God which is at Corinth, to those who are sanctified in Christ Jesus, called to be saints, with all who in every place call on the name of Jesus Christ our Lord, both theirs and ours:
Gal 1:1; Paul, an apostle (not from men nor through man, but through Jesus Christ and God the Father who raised Him from the dead),
Gal 1:2; and all the brethren who are with me, To the churches of Galatia:
Eph 1:1; Paul, an apostle of Jesus Christ by the will of God, To the saints who are in Ephesus, and faithful in Christ Jesus:
Php 1:1; Paul and Timothy, bondservants of Jesus Christ, To all the saints in Christ Jesus who are in Philippi, with the bishops and deacons:
Col 1:1; Paul, an apostle of Jesus Christ by the will of God, and Timothy our brother,
Col 1:2; To the saints and faithful brethren in Christ who are in Colosse:
1 Thess 1:1; Paul, Silvanus, and Timothy, To the church of the Thessalonians in God the Father and the Lord Jesus Christ:
Even parts of Romans are specifically for the church. ( Rom 1:7; )
Why would unbelievers read a letter from Paul?