How to build senior professional network in the US?

I have held the most senior engineering roles in some of the largest software companies in Australia and New Zealand. There has been a notable skill and experience gap between me and other engineers in these companies, which can be challenging at times. Before I get too old (maybe I already am!), I’d love to see if I could make it in the much larger and more competitive US scene. The problem is that I don’t really have a network in the US, given that we’re on the other side of the world.

Do you have any suggestions for how I could grow one? I’ve been reaching out on LinkedIn, but it just feels like cringe-worthy self-promotion and very superficial.