As to me? I don't know as I think despite being of Italian and French Canadian decent I was born to wander or somehow got Nomad blood.
Being with a large company isn't bad. We have about 25,000 people globally I believe. We also have several endless websites where positions within the organization are posted. Everything is done online. My resume is online as well as pages of information about my likes and dislikes. I actually didn't choose MDR (the company) but they sort of choose me when they bought the smaller company I worked for.
You have obviously been from wherever you are to the US and enjoyed your stay. I have no clue how long you were here. Remember something though, when you commit to "the big move" it is for years and not a visit. You really need to be sure you are ready to accept a new culture and way of living. For me that included learning the languages of my host countries. Additionally respecting and obeying their laws and traditions. You are no longer in your back yard but the back yard of another.
Ron