One idea is to read some of the insight publications that you find interesting and reach out to one of the authors. You'll have to have valid points to ask them about, and maybe some insights of your own that are impressive enough to warrant a follow-up email/call.
Note: I haven't actually tried this, but before I developed a network, I did consider doing this.
To give you a dose of reality, networking is hard. Even after I'd made a some internal BCG contacts, my application to attend their local experienced hire open house was rejected. My contact helped to get me into the event, which turned out to be critical to me getting an interview. My initial conversation with the recruiter when poorly, so I scheduled time with one of the consultants I'd met at the open house to discuss a shared interest/business idea, and I think he helped push my interview through.
To give you a bit of encouragement, networking does eventually pay off, and even distant connections help. My BCG referral actually came from a 1) friend's 2) mom's 3) former coworker's 4) BCG contact who'd reached out to #3 when looking for exit opportunities, who put me in touch with 5) another BCGer with interests similar to mine. The process took a few months. But even though #5 is based in a different city and I ultimately didn't get an offer from BCG, I will be meeting with her this week to continue discussing some of our shared business interests
It's totally a game, but it can be rewarding in the end. Keep at it!