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In Catholic theology, there are 4 "compartments" / "parts" of the underworld where souls are waiting in the Intermediate State (see Wikipedia article Limbo):
The Catholic Answer article Did the good thief go straight to heaven? leaves open two possible final destination of the thief:
CONCLUSION: Jesus definitely went to the Limbo of the Patriarchs to preach to the "spirits in prison". One interpretation is that "paradise" means this place, where Jesus's presence made the place a paradise. The thief thus went together with Jesus to the Limbo of the Patriarchs because at that point the gates of heaven would not have been opened yet until the resurrection. After resurrection, the thief may then go to heaven (or purgatory?) while Jesus came back to earth for 40 days (having been resurrected) to show himself to the apostles, to teach them, and to appear to hundreds more people before ascending to heaven.