I provide a lot of background, which may not all be essential. Feel free to skip to the question.
I was contacted by a woman who got a 3rd cousin match with me online. She told me that she was an adoptee curious about her biological father, although she did not want to intrude on his life. She told me the little she knew about her father: his ethnicity and that he was in a certain branch of the armed services at a certain base 9 months before she was born. (She had already determined the identity of her late biological mother and had enjoyed getting to know her relatives on her mother's side of the family.)
In our entire family tree, I only found one man of the right generation who had served in that branch of the armed services. He was a second cousin of my father (as would be predicted by the DNA match). Searching online records, I found that a few years before the woman's birth, he was stationed at the base she mentioned. I was confident that I had discovered her father. Let me add that few people of my ethnicity join the branch of the military of her biological father. (Her DNA results showed her to be approximately 50% of this ethnicity, which differed from her biological mother's.)
My father's second cousin is still alive. After getting his email address discreetly from a close relative of his (without saying why I wanted it), I gave the man the above information, emphasizing that the woman did not want to importune or embarrass him and that we were willing to keep the relationship secret. I asked him to let me know if my conclusion was incorrect, in which case we'd keep looking, or how he would like to proceed if he was the father.
A week later, I received a reply emphatically stating that he could not be the woman's father, that he had only one serious relationship in his life, with the woman he married. (I had known he had gotten married a year or two after the adoptee was born and that his wife was still living.) He said the adoptee would have to look elsewhere and that he wished her well with the search.
He did not say that he was not at that location at the time the pregnancy originated or that he did not know the biological mother, whose name I provided. Yesterday, I thanked him for the information, asking if he knew any other family member who was in that branch of the armed services in that location at that time. If so, would he please tell the man or me. I have not gotten a reply (and don't expect one).
I would still like to help this woman find the identity of her father (if she isn't too discouraged by what has happened so far). Of course, I will suggest comparing her DNA with those of others of my paternal relatives so we can narrow the match down. (Right now, her DNA is on a different site than most of my relatives. FYI, I am knowledgeable about genetic genealogy. I'm a longtime member of Genealogy Stack Exchange but am using a different account to protect people's privacy, even though I've obscured the details.)
How can I investigate this further without outing the father? If I email members of my extended family asking "Do you know anyone who was in <branch of the armed services>, serving at <name of base> in <year>?", everyone will think of this man (or anyone else in the family who meets the criteria). This would out the father, at least to relatives close enough to him to know these details. Neither the adoptee nor I wants to embarrass the father.
If I decide to start asking relatives, should I first let the veteran I emailed know that I will be doing so if he is unable to help me identify the father discreetly, or would that seem like blackmail? Should I wait to pursue this research until his death (he's an old man)? His wife's death? He has legitimate children.
Of course, I will do nothing without the adoptee's consent. What I would like to know is what course of action to offer/advise to her.
The adoptee does not want the matter pursued, and I will respect her wishes.