There is no 9–6 overlap. Stretching to 07:00–21:00 on both sides gives you 07:00–10:30 Chicago time:
07:00 you → 17:30 Hyderabad
08:00 you → 18:30 Hyderabad
09:00 you → 19:30 Hyderabad
09:00→19:30outside Hyderabad working hours
10:00→20:30outside Hyderabad working hours
11:00→21:30outside Hyderabad working hours
12:00→22:30outside Hyderabad working hours
13:00→23:30outside Hyderabad working hours
14:00→00:30next day · outside Hyderabad working hours
15:00→01:30next day · outside Hyderabad working hours
16:00→02:30next day · outside Hyderabad working hours
17:00→03:30next day · outside Hyderabad working hours
Working hours assumed 09:00–18:00 local. Offsets include daylight saving where it applies, so this shifts through the year.