Third, Acts never states that Paul dies at the end of Roman custody. In fact, Paul himself believed that he would beat his charge and get out of Roman imprisonment (Phil. 1:19; 25). Therefore, Paul was probably released from house arrest, and then, he continued to preach. Clement (an early second century writer) said that Paul went “to the extreme limit of the west” (1 Clement 5), which would have meant Spain. The Muratorian Canon (AD 200) states that Paul travelled to Spain. Moreover, Paul himself stated that he intended to preach in Spain, if he was allowed (Rom. 15:24). Eusebius writes:
Paul is said, after having defended himself, to have set forth again upon the ministry of preaching, and to have entered the city [Rome] a second time, and to have ended his life by martyrdom. Whilst then a prisoner, he wrote the Second Epistle to Timothy, in which he both mentions his first defence, and his impending death.
In addition, there are also a list of places that Paul mentions that aren’t recorded in the book of Acts. For instance, he mentions Spain (Rom. 15:24), Crete (Titus 1:5), Miletus (2 Tim. 4:20), Colossae (Philemon 22), and Nicopolis (Titus 3:12). These would all be best explained by a four missionary journey. For these reasons, it shouldn’t surprise us to see Paul mentioning details that are not recorded in the book of Acts, because these events probably occurred after his Roman house arrest. Therefore, these events in the Pastorals most likely occurred after the end of the book of Acts.
ARGUMENT #2: The Pastoral Epistles mention church leadership. Critics argue that official church leadership like this didn’t occur until the 2nd century.
 Cited in Earle, R. 1 Timothy. In F. E. Gaebelein (Ed.), The Expositor’s Bible Commentary, Volume 11: Ephesians through Philemon (F. E. Gaebelein, Ed.) Grand Rapids, MI: Zondervan Publishing House. 1981. 341.
 Ecclesiastical History (A. D. 326) 2:22. Cited in Earle, R. 1 Timothy. In F. E. Gaebelein (Ed.), The Expositor’s Bible Commentary, Volume 11: Ephesians through Philemon (F. E. Gaebelein, Ed.) Grand Rapids, MI: Zondervan Publishing House. 1981. 342.