As an interesting item that I came across . . .
Irenaeus possibly gives evidence that he thought that 2 Timothy was written during (one of ?) Paul’s Roman imprisonment(s). He notes that the first bishop of Rome after the apostles was Linus: ‘Paul mentions this Linus in the letters to Timothy’ (AH 3.3.3).
Linus, along with Prudens, Claudia, and all the brothers and sisters serve as those who greet Timothy (2 Tim 4.21).
It would appear that Irenaeus thinks this is a subsequent imprisonment to the one recorded in Acts 28. [Update: I'll pick up the reasoning for this in a second post.]