While the stone is still travelling on the elevator, there are two forces acting on it, the force from the elevator to the stone, as well as the weight due to gravity.
The moment the stone leaves the elevator, it becomes a free falling object. The elevator stops giving a force to the stone, and the only force remaining is its weight due to gravity.
From this you can see that as the only force is W = mg, the acceleration felt by the stone will be g.
While it is true it will be travelling upwards initially due to its momentum, its initial speed does not matter, as the only force that is acting on it would be force due to gravity, so its acceleration experienced will simply be g.