Rejected - email response

I was rejected for a role that I felt I was not only an excellent fit for, but I also thought that the interview with the head of the group went very well. I promptly sent email thank yous, and the recruiter responded saying thank you, but that I was not selected to move forward in the process. The email said that they cannot provide feedback per their company's policies. I wrote back quickly just saying that I was disappointed and thanked her for letting me know (no-response, and I didnt expect one). Now what do I write to the hiring manager? I honestly thought the interview went really well. She said things like, "your team" and also asked me who else I've met with on the team, then said she would talk to recruiting about next steps. Do I just simply say again I was disappointed and leave it at that or do I press for feedback? UPDATE: Thanks everyone for your feedback. I immediately responded to the recruiter saying thank you for the opportunity on Friday when I was released. I wrote the hiring manager yesterday a similar note. Here's what I said, if anyone ever needs a nice note. Hi [name], I learned from [name] that I would not be moving forward in the interview process and I wanted to reiterate that it was a pleasure to speak with you. Thank you again for your time and consideration. I thoroughly enjoyed meeting you and learning more about [company] and [division]. Please keep me in mind for future job openings where my skill set aligns. I wish you and the rest of the team at [company] all the best moving forward. My sincere regards, [name] She wrote me back right away, "Thanks very much for the note. Thanks for your interest and time. Best of luck to you and Happy Holidays."