Follow up for Unsolicited Application after just one Coffee Chat

I recently had a short coffee chat with an engineer at Company X that I had never met before. We talked about the company culture and his experience, and I introduced my background to ask his advice on exactly which skills I should reinforce to be useful to them.

During the chat, I mentioned that two months ago I applied for a position there but didn't get past the CV check. I told him I wasn't sure if I was rejected because my CV was badly written or if I just didn't have the right skills. He told me that it's hard to know how to pass their CV screening. He said getting past that initial check is hard, and they receive a huge number of applicants.

I still want to work at Company X, but I don't want to reapply for the role I already applied for (which is actually still open). I don't think I am the best fit for it, and I'd probably be desk-rejected anyway since my last application was so recent. Instead, I'd like to try an unsolicited application.

My questions:

• Is it appropriate to ask him to make a referral for an unsolicited application after just one coffee chat? Edit: I think it'd be better to ask for the email address of e.g. the CTO or of an engineer higher up in the hierarchy, and the possibility to mention our chat in the email

• If you think that is appropriate, do you have any tips on what exactly I should ask him to avoid friction?

By asking someone to refer you for a role, you are asking them to stake a small amount of their reputation on you.

By asking someone to refer you for a role which doesn't exist, you are asking them to stake a rather large amount of their reputation on you: "I think Leo is so good we should hire him even though you we don't have a role open right now".

Not going to happen after one coffee chat I'm afraid. Honestly, if someone asked me that I would be disinclined to interact with them again because it's pretty clear they were in it purely to use me as a stepping stone.