Firstly, the second death implies that it is after the first resurrection which is for the prophets and therefore there will be a second resurrection for the other believers and death for the unbelieving.
Secondly, there is no immortality of the Spirit before resurrection, ONLY GOD has immortality.
1Ti 6:16 Who only hath immortality, dwelling in the light which no man can approach unto; whom no man hath seen, nor can see: to whom be honour and power everlasting. Amen.
Rev 20:6 Blessed and holy is he that hath part in the first resurrection: on such the second death hath no power, but they shall be priests of God and of Christ, and shall reign with him a thousand years. This is the first resurrection
5 But the rest of the dead lived not again until the thousand years were finished.
Rev 21:8 But the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death.