Does the time difference between Chicago and Manila change during the year?

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

When is the best time to call Manila from Chicago?

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

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

Chicago

United States

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Manila

Philippines

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

City	Time zone	UTC offset
Chicago · United States	America/Chicago	UTC−5
Manila · Philippines	Asia/Manila	UTC+8

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