First, I recommend using E-mail for this, not the phone. E-mail allows the hiring manager to respond at her own convenience and to put some thought into her response. Say something like, "I appreciate your time speaking with me about the position, and I hope you'll keep me in mind if something opens up that you think would be a good fit. I also wonder if you might be willing to give me some advice for the future: Are there things you could share with me about what I could have done to be a stronger candidate and that I could do to be a stronger candidate in the future?"
When you say this, you must be absolutely clear in your own mind that this is not about debating the decision or trying to reverse it. If the hiring manager picks up on even a whiff of argumentativeness, she's going to revert to vague pabulum, like: "We decided to go in a different direction." After all, the decision has been made; she doesn't have time or interest in arguing it with you.
I recently had a candidate do this perfectly. Shortly after receiving our rejection, he sent me this: "If I could impose so much as to ask a favor, I would greatly appreciate it if you could take a few minutes to offer some criticism on the way I have presented myself: résumé, cover letter, interview, etc. Perhaps, for instance, there are weaknesses I am presenting without addressing or I am laying poisonous Easter eggs without knowing it. Please understand that I'm asking for help here, not trying to rationalize or pick a fight."
It's just human nature to want to help this guy, and I ended up giving him a very candid answer.
That said, I do know that some hiring managers will never give rejected candidates feedback, for fear of saying something that will open them up to a lawsuit. There's nothing you can do about these people; they are muzzled by fear. But even if you encounter that, there's still no reason not to give it a shot, as long as you're not defensive about it and are prepared for an honest critique. While some won't answer you or will tell you something so vague as to be useless, others will tell you something helpful—or even point you in the direction of a job lead.
And if you get an answer, no matter what it is, remember to say "Thank you." When I take the time to help someone with feedback and get silence in return, I remember it.
Alison Green is chief of staff for a medium-sized nonprofit where she oversees day-to-day management of the staff as well as hiring, firing, and staff development. She is working with the ManagementCenterto coauthor a book on nonprofit management. Her writings have been published in the Washington Post, the New York Times, Maxim, and dozens of other newspapers. She blogs at Ask a Manager.