Seems to me that Jesus is talking about divorce being allowed because of the hardness of hearts. Notice the word fornication is a different word used in the Greek than adultery later in the same sentence. We know a bill of divorcement is allowed in the law because of the sinful hearts of people would cause uncleanness in a person(which was not so at the beginning of creation). This uncleanness(Crookedness/unfavorableness) could not have been adultery since the sentence for adultery is death. If we look at the word fornication it means idolatry in a figuratively way which is why it is not the same word as adultery later in the sentence. This seems to be the same as what the law of God says. And we know Jesus did not come to change it or preach against it.
Now we must remember this sentence is an translation which would have a bias from the translator of that time and culture. If you ask me then Matthew 19:9 is basically saying this: If a man put away (Divorce) his wife, and it be not because of any (uncleanliness/crookedness/unchastity) in her(not a justified reason) , and (she) then go and marry another, commits adultery, and the man that shall marry her is also committing adultery.
Matthew 19:9
And I say unto you, Whosoever shall put away his wife, except it be for fornication(Unchastity RSV version), and shall marry another, committeth adultery: and whoso marrieth her which is put away doth commit adultery.
Deuteronomy 24:1
When a man hath taken a wife, and married her, and it come to pass that she find no favour in his eyes, because he hath found some uncleanness in her: then let him write her a bill of divorcement, and give it in her hand, and send her out of his house. 2And when she is departed out of his house, she may go and be another man's wife. 3And if the latter husband hate her, and write her a bill of divorcement, and giveth it in her hand, and sendeth her out of his house; or if the latter husband die, which took her to be his wife; 4Her former husband, which sent her away, may not take her again to be his wife, after that she is defiled; for that is abomination before the LORD:
In the case of 1 Corinthians 7:10-16 it says let not the wife depart from her husband, but if she (by her own will) depart, let her remain unmarried, or be reconciled to her husband. If the husband or wife is a believer and the other is not but pleased to dwell with each other then do not apart. But if the unbelieving part depart, let them depart. It seems to imply that if a woman departs she should remain unmarried or return, but if the husband departs from her then she is no longer in such bondage which seems to imply she is free to marry another (which the law also says in Exodus 21:10-11 since he is no longer providing for her which means she is free to go).
Exodus 21:10-11
if he take him another wife; her food, her raiment, and her duty of marriage, shall he not diminish.11And if he do not these three unto her, then shall she go out free without money.
1 Corinthians 7:10
And unto the married I command, yet not I, but the Lord, Let not the wife depart from her husband: 11But and if she depart, let her remain unmarried, or be reconciled to her husband: and let not the husband put away his wife.12But to the rest speak I, not the Lord: If any brother hath a wife that believeth not, and she be pleased to dwell with him, let him not put her away. 13And the woman which hath an husband that believeth not, and if he be pleased to dwell with her, let her not leave him. 14For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband: else were your children unclean; but now are they holy. 15But if the unbelieving depart, let him depart. A brother or a sister is not under bondage in such cases: but God hath called us to peace. 16For what knowest thou, O wife, whether thou shalt save thy husband? or how knowest thou, O man, whether thou shalt save thy wife?
This is what I see the law and NT writings is talking about. My understanding is not the end all or be all but this seems to make sense to me.
