The soul already is immortal (does not die) but still it has a beginning and an end. The human, through an upright life can contribute spiritual substances to the soul body so that it becomes more refined.
Eventually, however, the human individual is able express Spirit directly and there is no longer a need for the soul body and its substance is reabsorbed into the greater system.
The second death?
I have always thought of Eternal Life as realising or becoming cosciously aware that we ARE Spirit.
Who would want the restrictions (eternally) of a human physical body?
Our birth is but a sleep and a forgetting…trailing clouds of glory do we come from God, who is our home -Wordsworth
If you listen to Beethoven's 3rd Symphony, second movement, you may hear 3 death processes after physical death - etheric, astral and mental.
Since it is a hero that has died, he manages to cast off the mental body as well. This may not occur with many humans who finish floating on the astral plane, unwilling to lose their contact with those still in the body.