Ask HN: Best organizations to network

I'm looking at changing careers (and possibly countries) from a mathematics teacher (physics undergraduate degree from 2014) to (almost) anything else.

I'm in a small town, however, without much in the way of connections outside that town. What are some of the best possible organizations to join in a bigger city that would allow me to network more? I know of general fraternities, such as the Freemasons, as well as volunteer groups such as Rotary Club and the Lion's Club, etc., but would those actually be effective to join for networking? Especially if I wanted to move and find a job outside of the United States? Any suggestions, basically, on the best ways to find people to network with, for someone who isn't in a big city and can't go to a lot of meetup-like things?