Is it a moose that became a detective ? Or a detective who searches for moose?
Everyone can buy ONE building during the buy phase at a location where they have a posse.
After all players have bought buildings or declined, the leftover space let's you buy a "leftover" building.
The rules never outright state that this is IN ADDITION to a building you purchased. (Thus being a way to buy 2 buildings in one buy phase.) But it makes sense that this is what it means (and that's how we played.) But my personal preference would be for just a little extra clarity in the rules via the word phrase "even if they already purchased one."