What if the question isn't, "What can this person do for me?" but, "What value can I bring to this person?" How Can I Make Your Life Easier will sell more copies than How Can I Attach Myself To You. The young artist meets the established artist and nervously blurts out, "Let me know if you ever need an opener!!". What if instead you offered: "Let me know if you ever need someone to sell your merch," or "I'd be happy to guitar tech/load gear/wrap cables in exchange for tickets to your show!" In my band SISTERS, I rarely considered the former offer beyond an awkward "Totally, will do!!", but if any single one of them had offered the latter, I would have jumped on it in a heartbeat, and I probably would have eventually asked them to open for us.
The producing world isn't so different from the artist world in this regard. I've let the words slip out of my mouth a few too many times: "I'd love to jump on any projects you're working on..............." I might as well have said "I've been tweaking my mixing template for two weeks straight and no one is hiring me." Now if a young producer asked me, "Can I tune and align any vocals for you?" I would hand them my computer.
If you demonstrate the value you're able and willing to bring and ask for little in return, you will be able to expand your network with people who will turn to you first when they need a favor.