I have already written up the last section of this teshuva, where Rav Moshe Feinstein used the behavior and character traits of the suspected mamzer as a final proof that he was not, in fact, a mamzer. In this summary I will focus on the bulk of the teshuva, where Rav Moshe tackles the essential halachic issues.
The question dealt with a woman who was civilly divorced in 1956 but only received her get in 1959. Later that year she married another man, and a few months later she bore a son. Clearly this son was not conceived after the second marriage. The new husband claimed that the child was his, and that the child had been concieved before the woman received her get. If that claim would hold, the child would be the product of an affair with an eishet ish, and thus would be a mamzer. The issues at hand focus on the trustworthiness of both the husband and the mother with regard to whether or not they can proclaim that their son is a mamzer.
The weightiest issue dealt with in the teshuva is that of yakir, the idea that the Torah recognizes a father's "recognition" of his children for halachic purposes. While we normally believe a father to say that a certain child is his, Rav Moshe notes that that only applies when he has raised the child and we have no reason to think otherwise. However, in this case it is clear that the child was concieved before he married the mother, and thus he does not have the right to declare his son a mamzer under the rubric of yakir. Rav Moshe claims that he does not even qualify as an eid echad to make such a claim.
However, Rambam seems to take the opposite view on this, claiming that a person can assert that someone is his firstborn for inheritance purposes, even if that son did not grow up with that father. Rav Moshe investigates this position and ultimately concludes that Rambam has two different takes on the law - one in the laws of inheritance and one in the laws of yuchsin (lineage). For inheritance purposes, Rambam would allow someone to make a claim about a child that he did not raise, while for lineage purposes a person cannot make such a claim with any degree of accepted credibility.
Taking that into account, Rav Moshe rules that we do not believe the new husband that he and the woman conceived the child before the divorce. This ruling is based on a variety of reason. One, we have no evidence that this woman was ever involved in a licentious relationship especially since for the three years since her civil divorce there was no indication that she was ever involved in a sexual affair. Two, even if there were witnesses that they were in seclusion with one another before her get was given, the gemara in Ketubot rules that we do not proclaim that a person is forbidden based on the knowledge that they had had yichud with someone. Three, if the only testimony that they had an affair is based on what the man says, then he would be incriminating himself, and we do not accept self-incriminating testimony.
With regard to any claims by the woman, we have less of a reason to believe her, since yakir only applies by the father and not the mother. As such, there is no halachically acceptable testimony that would establish that the child was conceived before the get was given, and thus Rav Moshe rules that he is not to be considered a mamzer.