You do not ask "can you refer me" at the end. If they were going to, now they will be turned off by your lack of tact. Rather, you can "hint" at one (step f below) by saying something along the lines of "This has been so helpful. Thank you so much for your time. I will definitely be apply in the coming weeks - do you have any advice for me?"
Make sure to also "play tag" (step e) and get put in touch with someone else (i.e. preventing a dead end).
Feel free to reach out to discuss a bit further (verbalizing is a bit better than just writing steps), but I recommend the following:
a) Reach out to people in your network
b) Reach out to people once removed from your network
c) Reach out to people with a similar backgorund to you (i.e. same alma mater, same historically underrepresented demographic i.e. gender, orientation, ethnicity, etc., same career switch, etc.)
d) Tailor a message to them specifically both showing interest in them and their journey and demonstrating that you have done your research and could be a valuable hire
e) Play "tag" across calls you get so that you can work your way towards the company/office/role you want
f) Never directly ask for a referral, but "hint" at needing one (this is nuanced and important...happy to talk through wording)