How did literature contemporaneous to the New Testament understand the "Second Death"?

OP: What documents during and before the first century use the expression "second death," and what did it mean? Did it reference (1) eternal torment; (2) permanent, literal death/annihilation; (3) temporary, corrective punishment; or (4) something else? (I'm asking specifically for extra-biblical sources during and before the first century, not sources after.)

Philo a contemporary who wrote and lived about the time of Christ describes a "second death" in contrast to an "ending death" with hope.

Perhaps some one will say he [Cain] should have been put to death at once; this is a human mode of reasoning, fit for one who does not consider the great tribunal of all for men look upon death as the extreme limit of all punishments, but in the view of the divine tribunal it is scarcely the beginning of them. On Rewards and Punishments

For Christians, the great tribunal would be the great white throne judgment, which precedes the decision about the second death.

And I saw a great white throne and the one sitting on it. The earth and sky fled from his presence, but they found no place to hide. Rev 20:11

which then leads to judgment and the second death

Then death and the grave were thrown into the lake of fire. This lake of fire is the second death. Rev 20:14

This sequence is likewise with Philo.

Since then the action of this man [Cain] was a novel one, it was necessary that a novel punishment should be devised for him; and what was it? That he should live continually dying, and that he should in a manner endure an undying and never ending death; for there are two kinds of death; the one that of being dead, which is either good or else a matter of indifference; the other that of dying, which is in every respect an evil; and the more protracted the dying the more intolerable the evil. -ibid-

Live continually dying, as in the lake of fire, without hope.

The word used for "eternal punishment" in Matthew 25:46 is kolasis [kovlasi"]. According to Bauer writers during the New Testament period used it only of temporal torture and conscious torment in the afterlife. No other idea for koine Greek is recognized. Moulton and Milligan can find only examples in papyrus where kolasis [kovlasi"] involves the person actually feeling the punishment. It is used elsewhere in the New Testament only in 1 John 4:18, which says fear has torment. -source-

About 150 years later, Irenaeus picks up this theme of judgment and eternal destiny.

And in the Apocalypse John saw this new [Jerusalem] descending upon the new earth.4773 For after the times of the kingdom, he says, “I saw a great white throne, and Him who sat upon it, from whose face the earth fled away, and the heavens; and there was no more place for them.”4774 And he sets forth, too, the things connected with the general resurrection and the judgment, mentioning “the dead, great and small.” “The sea,” he says, “gave up the dead which it had in it, and death and hell delivered up the dead that they contained; and the books were opened. Moreover,” he says, “the book of life was opened, and the dead were judged out of those things that were written in the books, according to their works; and death and hell were sent into the lake of fire, the second death.”4775 Now this is what is called Gehenna, which the Lord styled eternal fire.4776 “And if any one,” it is said, “was not found written in the book of life, he was sent into the lake of fire.” Irenaeus Against Heresies Book V Chapter XXXV

This quote from Irenaeus could not be any clearer that the lake of fire the second death is Gehenna.

Gehenna is found in the Gospels.

'And if thy foot may cause thee to stumble, cut it off; it is better for thee to enter into the life lame, than having the two feet to be cast to the gehenna, to the fire -- the unquenchable -- YLT Mark 9:45

In addition to Philo and Irenaeus, we find the same reference to judgment and the second death in the lake of fire with Tertullian, writing about the same time as the latter.

Accordingly, we get ourselves laughed at for proclaiming that God will one day judge the world. For, like us, the poets and philosophers set up a judgment-seat in the realms below. And if we threaten Gehenna, which is a reservoir of secret fire under the earth for purposes of punishment, we have in the same way derision heaped on us. Tertullian Apology Chapter XLVII

SUMMARY

The lake of fire the second death is termed Gehenna in the gospel accounts. This definition is found in Irenaeus. Further, Philo also writes of the great tribunal the great white throne judgment that may lead to eternal torment without hope in the lake of fire.

"Second Death" is a term first found in the Book of Revelation. But if this phrase is synonymous with "Lake of Fire" then there were indeed contemporary equivalents in the time of Jesus. The best known would be the Dead Sea or its spiritual equivalent.

Biblical Data: the Dead Sea

• Genesis 19:24, Then the Lord rained on Sodom and Gomor′rah brimstone and fire from the Lord out of heaven.

• Deut 29:22-23 says of the tribe which turns to other gods: "the Lord would single him out from all the tribes of Israel for calamity, in accordance with all the curses of the covenant written in this book of the law. And the generation to come, your children who rise up after you, and the foreigner who comes from a far land, would say, when they see the afflictions of that land and the sicknesses with which the Lord has made it sick—the whole land brimstone and salt, and a burnt-out waste, unsown, and growing nothing, where no grass can sprout, an overthrow like that of Sodom and Gomor′rah, Admah and Zeboi′im, which the Lord overthrew in his anger and wrath.

• Two other Old Testament passages which refer "fire and brimstone" as God's punishment (Psalms 11:6; Ezekiel 38:22).

1 and 2 Enoch

An important influence on the Book of Revelation may be the Book of Enoch (67) where a molten Sea appears as the place of punishment for evil spirits.

I saw that valley in which there was a great convulsion and a convulsion of the waters. And when all this took place, from that fiery molten metal and from the convulsion thereof in that place, there was produced a smell of sulphur, and it was connected with those waters, and that valley of the angels who had led (mankind) astray burned beneath that land. And through its valleys proceed streams of fire, where these angels are punished who had led astray those who dwell upon the earth.

-2 Enoch ch 10 is another instance. Here it is not a lake or a sea but a river:

And those men carried men to the northern region; and they showed me there a very frightful place; and all kinds of torture and torment are in that place, cruel darkness and lightless gloom. And there is no light there, and a black fire blazes up perpetually, with a river of fire that comes out over the whole place, fire here, freezing ice there, and it dries up and it freezes; and very cruel places of detention and dark and merciless angels, carrying instruments of atrocities torturing without pity. And I said, “Woe, woe! How very frightful this place is!” And those men said to me, “This place, Enoch, has been prepared for those who do not glorify God, who practice on the earth the sin which is against nature, which is child corruption in the anus in the manner of Sodom, of witchcraft, enchantments, divinations, trafficking with demons, who boast about their evil deeds – stealing, lying, insulting, coveting, resentment, fornication, murder – and who steal the souls of men secretly, seizing the poor by the throat, taking away their possessions, enriching themselves from the possessions of others, defrauding them; who, when they are able to provide sustenance, bring about the death of the hungry by starvation; and, when they are able to provide clothing, take away the last garment of the naked; •who do not acknowledge their Creator, but bow down to idols which have no souls, which can neither see nor hear, vain gods; constructing images, and bowing down to vile things made by hands – for all these this place has been prepared as an eternal reward.

The above represent several examples of Jewish literature (some consider 2 Enoch to be Jewish-Christian) which help us understand the idea of the "Second Death" as described in the Book of Revelation: a place of eternal punishment after one's "first" or normal death.