1 Cor. 7:11 is the lynchpin to this whole conversation. Is there a duty on the husband to take back a wife who left?
The verse says she is to remain "unmarried". If there is no longer a marriage according to scripture, I don't know if you can bind duties upon the exhusband such as Ex. 21:10 in order to give a reason for a legitimate divorce. Unless you can show the husband has a duty to take back a departed wife, I don't see any authority for her to marry someone else.
There is precedent for the first husband to take her back. It’s called the parable of the unforgiving debtor. If Jesus was able to wash him clean by his precious blood then he has to forgive others. Therefore if she apologizes and 100% repented, then he should take her back as an additional, submissive, and obedient Christian wife. Only way I see, per scripture (if she repented), he can righteously refuse is if she’s been one flesh with another man - then God says it would be an abomination for him to take her back:
Deuteronomy 24:1-4
Suppose a man marries a woman but she does not please him. Having discovered something unlcean (Hebrew defines this word as something sexual - like public nakedness/exposure) about her, he writes a document of divorce, hands it to her, and sends her away from his house.
2When she leaves his house, she is free to marry another man.
3 But if the second husband also turns against her, writes a document of divorce, hands it to her, and sends her away, or if he dies,
4the first husband may not marry her again, for she has been defiled. That would be detestable to the LORD. You must not bring guilt upon the land the LORD your God is giving you as a special possession.
I know the reason for the divorce is different than the passage above, but I don’t think it changes the reasoning it would be detestable to the LORD for him to take her back. If she hasn’t been one flesh with any other man - I agree she should try and work harder to get the first husband to take her back. Pray to King Jesus to open his heart to forgiveness and pray you two can come back together. That would be the best outcome.