Failure at the Bay of Pigs did not deter the Kennedy administration from seeking ways to topple Castro. In November 1961 Operation Mongoose was created to coordinate these efforts. This was not a strictly CIA operation - General Edward Lansdale was tasked with coordinating activities between the CIA, Defense Department, and State Department. The Special Group Augmented (augmented by the President's brother Robert Kennedy and General Maxwell Taylor) oversaw the program at the White House, and Robert Kennedy played a very active role.
Mongoose entailed a wide range of activities, including intelligence collection, sabotage operations, searching for leaders within Cuba who could overthrow Castro, and more. The Northwoods operation, which contemplated faked and real terrorist activities which could be blamed on Castro and used as an provocation for invasion, were developed in this period with Lansdale's involvement.
Whether plots to assassinate Castro were part of this operation, and whether Robert Kennedy or President Kennedy condoned them, has remained a point of controversy to this day. Lansdale himself said that Robert Kennedy was aware of assassination plotting. One memo from Lansdale to RFK in early 1962, uncovered by the Church Committee, says that "we might uncork the touchdown play independently of the institutional program we are spurring."
The placement of Soviet missiles in Cuba, discovered in the fall of 1962, caused a suspension of Mongoose activities, and efforts to deal with Cuba took a different turn.