The biblical basis for saying that we won't have a second chance to be saved after we die is that the righteously default condition of every person before God is condemnation for having rejected the light that they were given:
For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life. For God sent not his Son into the world to condemn the world; but that the world through him might be saved. He that believeth on him is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God. And this is the condemnation, that light is come into the world, and men loved darkness rather than light, because their deeds were evil. For every one that doeth evil hateth the light, neither cometh to the light, lest his deeds should be reproved. But he that doeth truth cometh to the light, that his deeds may be made manifest, that they are wrought in God. - John 3:16-21
Jesus Christ, the Son of God, came into a world that is already under the just condemnation of God in order that those who believe might be saved through him. This sacrifice is pure mercy and love which God was under no judicial imperative to provide. It would not have been judicially "wrong" for God to consummate His righteous judgement of the condemnation of all rather than institute a means of salvation because all have actively sinned.
As it is written, There is none righteous, no, not one: There is none that understandeth, there is none that seeketh after God. They are all gone out of the way, they are together become unprofitable; there is none that doeth good, no, not one.
Their throat is an open sepulchre; with their tongues they have used deceit; the poison of asps is under their lips: Whose mouth is full of cursing and bitterness: Their feet are swift to shed blood: Destruction and misery are in their ways: And the way of peace have they not known: There is no fear of God before their eyes. Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God. - Romans 3:10-19
- Note: The condition of those who die unborn and those before an "age of accountability" are a different matter as these are usually described as "covered" rather than "in need of covering".
In order to remain Just and also justify the ungodly God sent His Son to die for them. This does not indicate Injustice prior to the advent of Christ, prior to the coming of the promised Messiah. It is not unjust for God to condemn but, apart from Christ, it would be unjust for God to forgive:
For all have sinned, and come short of the glory of God; Being justified freely by his grace through the redemption that is in Christ Jesus: Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God; To declare, I say, at this time his righteousness: that he might be just, and the justifier of him which believeth in Jesus. Where is boasting then? It is excluded. By what law? of works? Nay: but by the law of faith. Therefore we conclude that a man is justified by faith without the deeds of the law. - Romans 3:23-28
There is no sense in which some vast multitude of long ago had no light of truth to guide them unto salvation by grace through faith and will receive a first chance to believe after death. The Bible is clear that creation itself testifies of God's glory and that God's moral law is written in the hearts of all, even those who do not have the formal law:
For as many as have sinned without law shall also perish without law: and as many as have sinned in the law shall be judged by the law; (For not the hearers of the law are just before God, but the doers of the law shall be justified. For when the Gentiles, which have not the law, do by nature the things contained in the law, these, having not the law, are a law unto themselves: Which shew the work of the law written in their hearts, their conscience also bearing witness, and their thoughts the mean while accusing or else excusing one another;) In the day when God shall judge the secrets of men by Jesus Christ according to my gospel. - Romans 2:12-16
When we respond in faith to what light (of truth) we have been given we are given more light. It is not for us to judge that the lost pygmy tribe in the jungles of the Amazon have never received any light when Scripture flat out says that they have. Those who have been inwardly responding to the light they have will respond to the greater light of Christ also:
Thy mercy, O LORD, is in the heavens; and thy faithfulness reacheth unto the clouds. Thy righteousness is like the great mountains; thy judgments are a great deep: O LORD, thou preservest man and beast. How excellent is thy lovingkindness, O God! therefore the children of men put their trust under the shadow of thy wings. They shall be abundantly satisfied with the fatness of thy house; and thou shalt make them drink of the river of thy pleasures. For with thee is the fountain of life: in thy light shall we see light. - Psalm 36:5-9
Old testament saints, prior to the advent of Christ, lived a life of faith looking forward to the fulfillment of God's promise of a savior, first given to Adam and Eve in the garden. Abraham saw Jesus' day and was glad. They had light enough to believe and to be justified by faith in the Savior who was to come. Abraham believed and that belief was credited to his account before God as righteousness:
After these things the word of the LORD came unto Abram in a vision, saying, Fear not, Abram: I am thy shield, and thy exceeding great reward. And Abram said, Lord GOD, what wilt thou give me, seeing I go childless, and the steward of my house is this Eliezer of Damascus? And Abram said, Behold, to me thou hast given no seed: and, lo, one born in my house is mine heir. And, behold, the word of the LORD came unto him, saying, This shall not be thine heir; but he that shall come forth out of thine own bowels shall be thine heir. And he brought him forth abroad, and said, Look now toward heaven, and tell the stars, if thou be able to number them: and he said unto him, So shall thy seed be. And he believed in the LORD; and he counted it to him for righteousness. - Genesis 15:1-6
It can easily be deduced from Scripture that each moment of life, during which creation continually declares the Glory of God, constitutes yet another chance to respond in faith to light given. The condemnation of John 3 (quoted above) is not condemnation in the absence of light but in the active rejection, the hating, of light. This is why the call goes out that now is the day of salvation:
While it is said, To day if ye will hear his voice, harden not your hearts, as in the provocation. For some, when they had heard, did provoke: howbeit not all that came out of Egypt by Moses. But with whom was he grieved forty years? was it not with them that had sinned, whose carcases fell in the wilderness? And to whom sware he that they should not enter into his rest, but to them that believed not? So we see that they could not enter in because of unbelief. - Hebrews 3:15-19
The notion of a second chance after death for everyone turns the gospel message into something like, "Believe on the Lord Jesus Christ and you will be saved but you don't have to believe now because you'll get another chance later." Obviously this is entirely foreign to the entirety of the Biblical text.
Additionally, if there is a second chance after death wherein every person who ever lived gets another shot at believing in Jesus and, as some suppose, most or even all will believe because they will see His glory without veil, then this changes the biblical definitions both of faith and of salvation. Since salvation is clearly through faith, it falls upon those who hold to a second chance at salvation after death to demonstrate that faith is possible after death.
For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: - Ephesians 2:8
For we are saved by hope: but hope that is seen is not hope: for what a man seeth, why doth he yet hope for? But if we hope for that we see not, then do we with patience wait for it. - Romans 8:24-25
Now faith is the substance of things hoped for, the evidence of things not seen.
When Jesus returns in all His glory and every eye sees Him will this be saving faith according to the above definitions? Does saving faith wail?
Behold, he cometh with clouds; and every eye shall see him, and they also which pierced him: and all kindreds of the earth shall wail because of him. Even so, Amen. - Revelation 1:7