Asking mentor for resume push?

Hi all, at this stage I should 100% know how to do this already, but here I am.

I just finished my junior year summer internship at a Fortune 1000 firm, and I managed to meet 1-on-1 with the CEO to discuss my future plans after senior year (if I would return full time at his firm, do something else, go to b-school, etc.) I told him I'm very interested in IB, and he instantly told me while he didn't have personal connections to any senior bankers, he'd be more than willing to take my resume and pass it on to the banks that the firm works with (he told me mostly BBs with some EBs occasionally here and there). He also made it clear, "Let me know if there's anything else I can help you with," etc etc.

I initially sent him my resume at the time, which didn't have the junior summer internship on there. Now that I've finalized it with what I did during the summer, I want to send it to him again with the hopes that he pushes my resume to the banks the firm has connections to. How do I phrase this question? Here's the ideas I have now, which I'm sure are not a good way to phrase it:

"Earlier I sent you a preliminary draft of my resume, but now that I have finalized it, I'd like to send it to you [what the fuck do I put after this?]"

"I understand you are very busy, but would you... [something about asking for advice / something not directly asking him for a resume push??]?

Thanks in advance for reading.