What we often mean by the common use of the term “hell” is a place of torment for unbelievers. The final place of eternal torment for unbelievers is the lake of fire (Rev 20:10), but they will not be sent there by God until his final act of judgment known as the Great White Throne judgment (Rev 20:11-15). That will be a time of bodily, spiritual torment.
The parable of Lazarus and the rich man in Luke 16:19-31, is often cited to describe the current state of unbelievers who have died, although there is debate as to whether or not the details given in the parable were meant to be taken as literal depictions of the afterlife. Many commentators argue that the main point is that Lazarus is in the place of the righteous and the rich man is in the place of torment, and nothing can bridge that great divide; they do not take it to mean that Jesus is describing the actual state of souls now. In support of this parable, though, are Jesus’ words to the thief on the cross (Lk 23:43) which indicate he will be with Jesus that day in paradise and Paul’s words in 2 Cor 5:8 which imply that when believers die they go immediately into the Lord’s presence.
Most of the time when eternal punishment or heaven are discussed in the scriptures the final judgment is in view, but these passages do not seem to indicate that. Even now, when people die they either go to be with the Lord or they go to a place of conscious torment.
That being said, I want to affirm that your understanding of the situation is correct. When an unbeliever dies at this time, his soul is separated from his body. The biblical evidence cited above does seem to indicate that believers go to a place of joy with the Lord and unbelievers go to a place of torment. At the final judgment of God, at the end of Christ’s Millennial reign on earth, all unbelievers will be united with their bodies and then sent to the lake of fire for all eternity.