Firstly: 

The verse (interpretation of the meaning) “The adulterer marries not but an adulteress...” [an-Noor 24:3] does not mean that the fornicator or adulterer cannot marry anybody but a fornicatress or an adulteress, or that the fornicatress or adulteress cannot marry anybody but a fornicator or adulterer. Rather what the verse means is that it is haraam for a man or woman who has committed zina to marry one who is chaste, unless they repent. 

Ibn Katheer (may Allah have mercy on him) said: Imam Ahmad ibn Hanbal (may Allah have mercy on him) was of the view that the marriage of a chaste man to an unchaste woman is not valid so long as she remains like that, unless she is asked to repent. If she repents, the marriage contract will be valid, otherwise it will not. Similarly, it is not valid for a free, chaste woman to be given in marriage to a man who is immoral and unchaste, unless he repents sincerely, because Allah, may He be exalted, says (interpretation of the meaning): “Such a thing is forbidden to the believers” [an-Noor 24:3]. End quote. 

Shaykh ‘Abd ar-Rahmaan as-Sa‘di (may Allah have mercy on him) said: What this verse means is that if a person commits zina, man or woman, and has not repented from that, then the one who wants to marry him or her even though Allah has forbidden that, must be:

- either one who does not adhere to the rulings of Allah and His Messenger, and such a person cannot be anything other than a mushrik;

- or if he does adhere to the rulings of Allah and His Messenger, but he wants to marry this person even though he is aware of her zina, then this marriage itself is zina, and the one who wants to get married is an adulterer and unchaste. If he truly believed in Allah, he would not want to do that.

This clearly indicates that it is haraam to marry a fornicatress or adulteress unless she repents, and it is haraam to marry a fornicator or adulterer unless he repents. 

End quote from Tayseer al-Kareem ar-Rahmaan fi Tafseer Kalaam al-Mannaan (p. 561). 

Therefore, it is not permissible for a chaste man to marry a fornicatress, and by the same token it is not permissible for a chaste woman to marry a fornicator, unless the one who has committed zina repents. Similarly, it is not permissible for the man or woman who has committed zina to get married unless they repent. 

For more information, please see the answer to questions no. 122639 and 14381.

Secondly: 

If a person repents from zina, he is no longer described as a fornicator or adulterer, therefore the prohibition on marriage of a fornicator or adulterer no longer applies to him; in that case it is permissible for him to marry a chaste woman, i.e., one who never committed zina; it is also permissible for him to marry a woman who committed zina previously but has now repented. 

The same applies to the woman who has committed zina. She does not have the right to marry a chaste Muslim man unless she repents. But if she repents, it is valid for her to marry him. 

With regard to what you have mentioned about the fornicator who has been given a hadd punishment not being allowed to marry anyone but a woman who is like him, who committed zina and has been subjected to the hadd punishment, this is a view that was narrated from some of the early generations, for which they quoted as evidence the report narrated by Abu Dawood (2052) from Abu Hurayrah (may Allah be pleased with him) who said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “The fornicator who has been flogged may not marry anyone but a woman who is like him.”

It was classed as saheeh by Shaykh al-Albaani in Saheeh Sunan Abi Dawood. 

Al-Qurtubi (may Allah have mercy on him) said: az-Zajjaaj and others narrated from al-Hasan that he said: What is meant is the fornicator and fornicatress on whom the hadd punishment has been carried out. He said: This is the ruling from Allah, so it is not permissible for the fornicator on whom the hadd punishment has been carried out to marry any but a woman on whom the hadd punishment has been carried out. Ibraaheem an-Nakha‘i said something similar. 

In Musannaf Abi Dawood it is narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “No fornicator on whom the hadd punishment has been carried out should marry anyone but one who is like him.” And it was narrated that one on whom the hadd punishment had been carried out (because of fornication) married one on whom it has not been carried out (i.e., one who was chaste), and ‘Ali (may Allah be pleased with him) separated them. 

The correct view is that the description “one who has been flogged” in the hadeeth refers to one who is known to have committed zina and it has been proven against him, and that can only apply to one on whom the hadd punishment has been carried out. Therefore what the hadeeth means is: it is not permissible for a chaste woman to marry one who is known to have committed zina, and it is not permissible for a man to marry a woman who is known to have  committed zina. So the meaning of the hadeeth is in accordance with the meaning of the verse (interpretation of the meaning) “The adulterer marries not but an adulteress...” [an-Noor 24:3], and reinforces that meaning. 

Ash-Shawkaani (may Allah have mercy on him) said: The words “the fornicator who has been flogged” refer to one who is known to have committed zina. This indicates that it is not permissible for a woman to marry a man who is known to have committed zina; similarly, it is not permissible for a man to marry a woman who is known to have committed zina. This is indicated by the verse mentioned in the book, because at the end of it it says“Such a thing is forbidden to the believers” [an-Noor 24:3]. End quote from Nayl al-Awtaar, 6/201 

Based on the above, if the woman regrets what she fell into of fornication and repented from it before that man married her, then the marriage contract is valid. But she has to conceal her (past misconduct) and not tell anyone about what she did previously. 

But if the marriage contract was done before she repented from zina, then the opinion on which fatwas on this website are based is that the marriage is not valid and the marriage contract must be re-done. See the answer to question no. 85335

Therefore, if it is possible to repeat the marriage contract – if the marriage contract was done before repentance – even if that is with any acceptable excuse, then this is what should be done and is more on the safe side, so as to avoid a matter concerning which the scholars (may Allah have mercy on them) differed, and it is more on the safe side for the marriage contract. 

But it is not possible to do that except by stating clearly that zina occurred, and if doing so will lead to negative consequences, such as if the husband will divorce the wife if he finds out about her past, or at least it would create mistrust and doubt on the part of the husband if he agrees to keep his wife with him, or it will disclose her fault among the people or cause her shame, and other negative consequences, then there is no blame on her, in sha Allah, if she continues with this marriage contract. Undoubtedly this opinion carries weight and has a valid foundation; in fact it is the view of the majority of scholars, especially with regard to one who entered into that marriage contract believing that it was valid. 

In fact some of the Hanbalis themselves even stated clearly that marriage in the case mentioned is valid. 

Al-Mirdaawi (may Allah have mercy on him) said: Some of our companions said: it is not haraam for her to get married before repentance if someone other than the fornicator marries her. This was stated by Abu Ya‘la as-Sagheer.

End quote from al-Insaaf, 8/133 

To sum up: 

If it is not possible to renew the marriage contract between the two families except by causing greater negative consequences, or creating a scandal and exposing the woman to shame, there is nothing wrong with her remaining in this marriage and letting her husband be intimate with her, and living her life in a normal manner. 

For more information, please see the answer to question no. 131467 

And Allah knows best.