Nigeria's main militant group, the Movement for the Emancipation of the Niger Delta, said it was behind the November attack in which seven Exxon workers were kidnapped and later rescued.
Oso is one of Nigeria's biggest condensate fields with about eight platforms whose total output averages about 75,000 bpd.
The attack by armed gang members prompted the U.S. energy firm to shut in 45,000 bpd of condensate and natural gas production, which is now beginning to come back onstream.
We have restarted 15,000 barrels per day of condensate from the Oso field. No firm time for restart of Natural Gas Liquids (NGL) production is available at this time, Exxon said in a statement.
Nigeria is Africa's largest oil and gas industry but years of attacks on pipelines and oil infrastructure has held back production and investment.
President Goodluck Jonathan brokered an amnesty with militants last August, leading to more than a year without major unrest but there has been a resurgence in violence in the last two months.
(Reporting by Joe Brock; editing by James Jukwey