Here's the thing: when I answer to such a question that I'm French, I also do get the followup question "where in France?". (although if the question is "where do you come from" rather than "where are you from", I now answer that I come from Japan, which gets WTF looks and different followup questions (but I only do that outside Japan, where I live))

This isn't an experience unique to people from the US (see other comments). Would you consider saying "State Name, USA"?

My mistake, I see now the person I responded to may have meant that Americans say just the state. In the videos I've watched, and at least for myself, I usually hear it like you suggested: state + USA.

Indeed, that's what I meant. And for less known states, that's usually followed by an awkward silence until they do say US.