What is the basis in the Scripture for the doctrine that those who haven't received salvation during their physical life, especially those who heard of Christ and yet chose not to pray to Him, will still have a chance to get saved after their physical death?
The only passage I am aware of is 1 Peter 3:18-20a:
For Christ died for sins once for all, the righteous for the unrighteous, to bring you to God. He was put to death in the body but made alive by the Spirit, through whom also he went and preached to the spirits in prison who disobeyed long ago when God waited patiently in the days of Noah while the ark was being built.
It should be noted that this is rather a "standalone" passage, in that there is nothing similar in Scripture, so it is unwise to base a doctrine on it. However, you could read into this:
- Jesus is here preaching to dead people in hell (implied by "he went and preached to the spirits in prison who disobeyed") and
- He would not preach unless there was hope of their salvation.
Note though that this is speculation and not a central belief of any mainstream Christian denomination (that I know of - but see @brillant's comments, below).
Then Judas assembled his army and went to the city of Adullam. As the seventh day was coming on, they purified themselves according to the custom, and kept the sabbath there.
On the next day, as had now become necessary, Judas and his men went to take up the bodies of the fallen and to bring them back to lie with their kindred in the sepulchres of their ancestors. Then under the tunic of each one of the dead they found sacred tokens of the idols of Jamnia, which the law forbids the Jews to wear. And it became clear to all that this was the reason these men had fallen. So they all blessed the ways of the Lord, the righteous judge, who reveals the things that are hidden; and they turned to supplication, praying that the sin that had been committed might be wholly blotted out. The noble Judas exhorted the people to keep themselves free from sin, for they had seen with their own eyes what had happened as the result of the sin of those who had fallen. He also took up a collection, man by man, to the amount of two thousand drachmas of silver, and sent it to Jerusalem to provide for a sin offering. In doing this he acted very well and honorably, taking account of the resurrection. For if he were not expecting that those who had fallen would rise again, it would have been superfluous and foolish to pray for the dead. But if he was looking to the splendid reward that is laid up for those who fall asleep in godliness, it was a holy and pious thought. Therefore he made atonement for the dead, so that they might be delivered from their sin.
Judas not only prayed for the dead men ("turned to supplication"), he also made a sin offering on their behalf. It is also the first reference of the resurrection in the scriptures.
Keep in mind that this is considered Scripture in Eastern Orthodoxy.
One more place in Bible which is (as far as I know) 1Corinthians 3:15. Here I post it with some context (1Cor 3, 10-15).
10 By the grace God has given me, I laid a foundation as an expert builder, and someone else is building on it. But each one should be careful how he builds. 11 For no one can lay any foundation other than the one already laid, which is Jesus Christ. 12 If any man builds on this foundation using gold, silver, costly stones, wood, hay or straw, 13 his work will be shown for what it is, because the Day will bring it to light. It will be revealed with fire, and the fire will test the quality of each man’s work. 14 If what he has built survives, he will receive his reward. 15 If it is burned up, he will suffer loss; he himself will be saved, but only as one escaping through the flames.
It definitely talks about those who build on Christ and will be saved, but it recognizes two categories: those who recieve reward and those who will be saved too, "but only as one escaping through the flames". Catholics interpret this as a reference to fate of those saved after death: they will go to heaven either directly (the reward) or through purgatory (the flames). Or through "something like purgatory" - I think Eastern Orthodox doctrine teaches something like this.