You will need to change orthographic size AND move the camera to compensate for the changed size. You will have to work out the position the player is in the current ortho size (camera.worldtoscreenposition) and then once you change the size, use screen to worldposition to work out where the player will need to be in the new camera, and move your camera appropriately.
I've not really played ski safari so I couldn't tell you how you'd get the same effect, but if you work out the above, your camera + player will be at the correct position, and then you can tweak it to feel more similar.