First, it must be recognized that HaShem hates divorce (Malachi 2:16), and reconciliation is preferred wherever possible (1 Cor 7:10-11; Matthew 18:15-17). “When a man puts aside the wife of his youth, even the very altar weeps” (TB, Gittin 90b). But, He does allow divorce in three circumstances:
- adultery (D’varim 24:1; Matthew 5:32; 19:9; Luke 16:18)
- abandonment by an unbelieving spouse (1 Corinthians 7:15)
- abuse (physical, emotional, spiritual) suffered by the wife (Shemot 21:10f)
In the case of  adultery, the innocent party is free to remarry (Matthew 5:32; 19:8-9; Luke 16:18), and also in  abandonment (D’varim 24:1-4; 1 Corinthians 7:15), providing a writ of divorce exists. If it does not, the abandonee is agunah (anchored) and cannot remarry.
Freedom to remarry is not in view in the case of  abuse, except that such be a reconciliation/remarriage with the former spouse (1 Corinthians 7:10-11). Remarriage to another party would thus only be possible after the ex-spouse is deceased (D’varim 20:7; Romans 7:1-3; 1 Corinthians 7:39). Once either party has remarried, reconciliation of the first marriage is off the table (D’varim 24:1-4), but as long as the ex-spouse is still living, the innocent spouse is considered unmarriageable.
Some see Avraham sending Hagar away as an instance of 3, i.e. divorcing her to protect her from abuse (B’reshith 16-21). It is not clear from the text, however, that a formal divorce is actually in view.
There is nowhere in Scripture a call or command to end the second marriage in divorce, even if it is not a biblical remarriage. The same 3 circumstances above would be the only grounds for termination (or annulment) of the marriage covenant, regardless of how it came to be entered into.