I've changed the text to better align with the way we've been playing it.
The way it's supposed to work is that you really shouldn't be Taken Out during a Solo. There's no, like, rule against it or anything, but it's much, much more likely that you'll be Taken Out during the Build-Up, and usually as a result of choosing to bump up your roll.
For example, in a playtest a PC jumped through a window to knock the gun out of the bounty's hand. This was in the Head Out during their Build-Up, and they'd already taken an injury during their Solo. So they ended up Taken Out and on the floor with glass sticking out of them for the Moment of Truth. It just made sense that the maneuver had really cost them something.
RAW, the person “closest” (whatever that means in context) to the bounty has narrative control, so they can choose whatever makes the most sense.
In practice, though, the most satisfying outcome is usually fairly obvious and we just talk it out at the table. Sometimes the whole crew works together, but sometimes the bounty has, like, a personal connection to one of the hunters.