It's complicated! The short answer I gave elsewhere: B cells aggressively replicate and mutate in response to pathogens, and those that match are selected for. Repeating this process gets more B cells, but also broader varieties of resultant antibodies matching not just the original pathogen but things that are close to it.
> Is there any way to know whether that increased breadth of antibody protection isn't coming from a post vaccination exposure that wasn't tested for not tested?
Yes. We collect plasma from people, and then test it against a variety of variants in assays. We have plenty of old plasma to do this with.
Indeed, there's some evidence that many people who were both infected with SARS-CoV-1 and subsequently vaccinated against SARS-CoV-2 may have exceptionally broad immunity to coronaviruses, not isolated to just SARS-CoV variants but also to wild-type coronaviruses circulating in animals and human coronaviruses causing the common cold.