West Texas Intermediate for January delivery, the front-month contract, was down 12 cents to $35.40 a barrel by 0248 GMT after finishing down nearly 5 percent on Wednesday.
Brent crude for February delivery, the front-month contract from Thursday, fell 21 cents to $37.18. The global benchmark fell $1.34 to $37.39 the previous session.
"Last night's inventory data from the U.S. was clearly unsettling," said Ric Spooner, chief market analyst at CMC Markets in Sydney.
"We are now seeing signs of the U.S. dollar getting stronger in our time zone as well, following on from the Fed," he said.
The dollar added almost 1 percent to 98.812 against a basket of major currencies and looked set for another test of stiff resistance around the 100.00 mark.
The U.S. Fed hiked interest rates for the first time in nearly a decade on Wednesday, a sign it believes that the U.S. economy had largely overcome the calamity that was the 2007-2009 financial crisis.
Higher U.S. rates typically support the dollar, making oil and other commodities denominated in the greenback more expensive, undermining demand.