Does the time difference between Chicago and Hong Kong change during the year?

Yes. Because of daylight saving time the gap moves between −14h and −13h.

When is the best time to call Hong Kong from Chicago?

There is no overlap between standard working days, so one side has to call early or late.

Chicago is 13h behind Hong Kong. This gap is not fixed: daylight saving time moves it between −14h and −13h during the year.

Chicago

United States

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Hong Kong

Hong Kong SAR China

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A 9-to-5 working day in Chicago does not overlap one in Hong Kong at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Chicago · United States	America/Chicago	UTC−5
Hong Kong · Hong Kong SAR China	Asia/Hong_Kong	UTC+8

What is the time difference between Chicago and Hong Kong?: Chicago is 13h behind Hong Kong. This gap is not fixed: daylight saving time moves it between −14h and −13h during the year.
Does the time difference between Chicago and Hong Kong change during the year?: Yes. Because of daylight saving time the gap moves between −14h and −13h.
When is the best time to call Hong Kong from Chicago?: There is no overlap between standard working days, so one side has to call early or late.