I think there's also something to be said to just "doing things" that are in the public eye.
What I mean by that is that (part of) the reason I got the job I'm at right now is because I happened to do something at an org that my boss worked at years prior. It wasn't a "connection" in the sense that we knew each other or ever crossed paths, but it still got me a foot in the door.
If you can make a meaningful contribution to a well known project or organization that will open doors via "connections" that you never knew existed.