The prime example I can think of is Russia. In 1917, the February Revolution put a coalition government, headed by Alexander Kerensky, in charge after Tsar Nicolas abdicated the throne. The October Revolution put Lenin and the communists in charge of the government although a civil war and unsuccessful counter revolutions went on until 1921.
Edit to add some additional ones, based on the criteria of less than 12 months between coups...
In Bolivia, on October 6, 1970 Juan José Torres seized power from Alfredo Ovando Candía. In turn, on August 21, 1971, General Hugo Banzer staged a successful coup with the help of the military and, perhaps, assistance as part of Operation Condor.
In Burundi in July 1966, Prince Ntare V deposed his father. On November 28th of 1966, Michel Micombero, who had earlier assisted Ntare, staged a coup, abolished the monarchy and took power himself.
In Ethiopia, September 12, 1974 Aman Mikael Andom overthrew Emperor Haile Selassie I. On November 17th 1974, Andom was killed by forces loyal to the Provisional Military Administrative Council (aka Derg).
In 1987, Fiji had a string of coups.
In 1983, the island of Grenada had several coups, finally ending with the intervention of the US.
Haiti saw several coups from 1988 onward after the departure of Jean-Claude "Baby Doc" Duvalier in 1986 up until the threat of US intervention in 1994. Leadership changed hands several times and exactly who was in charge wasn't always clear.
In 1994, the Rwandan Civil War resulted in several leadership changes and terrible genocide.
There are certainly some others I didn't cover here so feel free to add more examples.