There is no 9–6 overlap. Stretching to 07:00–21:00 on both sides gives you 07:00–11:00 Seoul time:
07:00 you → 17:00 Austin prev day
08:00 you → 18:00 Austin prev day
09:00 you → 19:00 Austin prev day
09:00→19:00prev day · outside Austin working hours
10:00→20:00prev day · outside Austin working hours
11:00→21:00prev day · outside Austin working hours
12:00→22:00prev day · outside Austin working hours
13:00→23:00prev day · outside Austin working hours
14:00→00:00outside Austin working hours
15:00→01:00outside Austin working hours
16:00→02:00outside Austin working hours
17:00→03:00outside Austin working hours
Working hours assumed 09:00–18:00 local. Offsets include daylight saving where it applies, so this shifts through the year.