Paul’s letters are the earliest Christian documents we have; he wrote during the church’s most formative years, when it still expected the Parousia to come at any moment, and we see in his letters that his thought about the afterlife progressed and matured as, contrary to his expectation, the first generation of Christians began to die. This watershed realization, that the Parousia would likely not happen before the first Christians—especially Paul himself—died, drove Paul to think more deeply about what exactly would transpire after someone dies.
Early in Paul’s career, his pareschatology is quite general: the dead in Christ are simply “asleep” until the Parousia, at which point they are “raised” or “awakened”—that is, resurrected—and God brings them with him down to the earth. In fact, Paul makes no direct mention of the intermediate state during this period; instead, we are left to infer an intermediate state through the gap in time between the believers’ deaths and the Parousia, which Paul, at this point, still expected to see in his lifetime.
Later in his career, Paul has a change of thought. He has realized that he will likely die before the eschaton, so he thinks through the nature of the intermediate state more fully. In discussing his own fate after death, he teaches that, after death, the person appears before Christ to be judged, “so that each may receive recompense for what has been done in the body, whether good or evil” (2 Cor 5:10). He expresses anxiety that he will be “found naked” and hopes instead that he will be “further clothed” with immortality (2 Cor 5:3-4). Ultimately, though, Paul is assured that he will be “with Christ” after death, because God has guaranteed it through the Holy Spirit and Christ has made Paul his own (2 Cor 5:5; Phil 1:21-23; 3:12). At this point in his career, Paul also clearly believes in an underworld where the dead await the Parousia; he cites a hymn or confession that teaches a three-story universe (heaven, earth, and underworld) and modifies a passage from Deuteronomy to discuss Jesus descending to the underworld (Phil 2:5-11; Rom 10:6-7).
Several themes are not specific to either his earlier or later periods. One such theme is that death is God’s enemy, which Jesus defeated through his resurrection (1 Cor 15:26, 54-55). Even when Paul talks about death as God’s punishment for sin, he always counters that notion with the good news that Jesus has conquered death. Throughout his career, Paul maintained that death does not separate the believer from Christ—that Christ, because he has conquered death, gives believers the power not to avoid death, but to go through death and survive it (Rom 7:24; 8:10, 13, 33-39; 1 Thess 5:9-10). Moreover, Christ, as the “the first fruits of those who have died” (1 Cor 15:20), is a signal that the rest of the dead would soon “ripen,” so to speak, and rise from the dead, as well. 
The value of the deutero-Pauline letters for this topic is that they provide clear evidence of Paul’s doctrine, as received by early Pauline Christianity. In these letters—specifically, Ephesians—the Pauline community believed that Jesus descended into the underworld after his death, which, along with Philippians 2:5-11 and Romans 10:6-7, is evidence that Paul taught the existence of an underworld where souls spend the intermediate state (Eph 4:9). This community also believed that the resurrection had already begun, in a spiritual sense, picking up on Paul’s discussion of the “spiritual body” in 1 Corinthians 15 (Eph 2:4-7). Thus, the deutero-Pauline letters serve as further proof that Paul taught an intermediate state.