From the USA
There is only one airline flying to American Samoa from the USA, Hawaiian Air. Hawaiian air has 2 flight from Honolulu, Hawaii to Pago Pago, American Samoa every week. Flights depart Honolulu on Mondays and Thursdays only. They also depart Pago Pagon for Honolulu every Monday and Thursday night. Keep in mind you will need your US Passport and round-trip ticket to enter.
From elsewhere
There are multiple daily flights from neighboring Samoa to American Samoa via Samoa Airways and Talofa Air. Flight times are 20 minutes on Talofa Air and 35 minutes on Samoa Airways.
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To get to neighboring Samoa, you can fly from Auckland, New Zealand; Brisbane or Sydney, Australia, or Nadi, Fiji.
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https://www.airnewzealand.com/
You may require an OK TO BOARD pass to enter depending on your country of origin or passport. More info can be found at https://www.touramericansamoa.com/entry-into-as