Is there a word for the situation where you have a conversation with someone and later on you think back on it and wish you would have said something that you didn't say?
I think German actually has a word for this and I would accept that.
Based on your comment:
where I wish I would have explained something about myself with a story, but I didn't think to do it at the time
For that specific situation, one might call it an Afterthought
An idea, response, or explanation that occurs to one after an event or decision
In addition, I think a good general phrase for that would be Missed Opportunity. Where you had a chance to add additional information but either forgot, or it didn't come up naturally in conversation.
In English, we use a French idiom esprit de l'escalier to refer to a witty retort that we could have made during an argument, but which only comes to us (often shortly) after the opportunity has passed.
Here's the definition from Oxford Dictionaries Online:
Used to refer to the fact that a witty remark or retort often comes to mind after the opportunity to make it has passed.
- Example sentences
‘I am frequently afflicted with esprit de l'escalier.’
‘Now if I'd managed to say all that off the cuff then I would be a genius comedian, but as it was I had to make do with experiencing esprit de l'escalier on the way home.’
A comeback is a retort made at the time.
An afterwit is the delayed kind of comeback which is only thought of after the event.
Afterwit : A good comeback, retort one thinks of only after the end of discussion or after leaving a social gathering.
There is also the term staircase wit which refers to the comment made by Denis Diderot (Wikipedia) after he was lost for words and only thought of a retort when he was 'at the bottom of the stairs'.
Not quite that exact meaning of specifically not having said something, but the phrases in hindsight or in retrospect spring to mind, i.e:
In hindsight, I should have mentioned I have a PhD.
In retrospect, I don't think boasting about my 'days without showering' record was such a good idea.
A relevant phrase is also "with the benefit of hindsight" which can be used similarly:
With the benefit of hindsight, I can now say that flirting with the interviewer was a bad idea.
Note. There is no equivalent "benefit of retrospection" phrase as far as I'm aware.
There's also a related concept called the hindsight bias (or as per the witty quote: "Hindsight is always twenty-twenty") --- this is when one judges in hindsight the outcome of a decision as being more predictable than it actually was when the decision was made.
Your comment about the job interview suggests that perhaps this isn't the tone you want, but a joke term usually associated with the sense of missed witty retorts and comebacks is "afterism", modelled on "aphorism".
aphorism, n.: A concise, clever statement.
afterism, n.: A concise, clever statement you don't think of until too late.
I've always liked this one. I came across it in a book called "Weird Words" years ago and it's one of the few words from the book I've remembered, though references to it are hard to find.
This post has a more thorough explanation of the origin and usage: https://stancarey.wordpress.com/2009/03/09/tintiddle-and-lesprit-de-lescalier/
It has been mentioned, but since it is not an actual answer so far, I am going to suggest the English equivalent of esprit d'escalier,
A few links to back that up:
Wiktionary defines it as "Thinking of an idea or course of action too late to use it effectively, or the tendency to do so" and notes that it is a translation of the French expression.
It also says "The French borrowing l'esprit de l'escalier is more often used, or occasionally the equivalent German calque Treppenwitz". But, as a German native speaker, Treppenwitz means something different to me, as well as to Duden, where it is defined as "Vorfall, der wie ein schlechter Scherz wirkt" ("an incident that seems like a bad joke"). It is heard most often, by far, in the expression "der Treppenwitz der Geschichte" (roughly, "History's bad joke").