Networking without feeling pushy (update)

I sent 6 messages on a professional network and got two replies. I want advice, not a job directly. How do you phrase the request?

Follow up with a thank-you and a short update. One of my chats led to a referral a month later.

Thanks for writing this out. I went through something close to it last year, and it got easier once I kept a simple log of every call and form.