Modifying the output layer of neural networks is part of transfer learning.
Adding new classes/output nodes always requires additional training.
One common option is just to retrain the last layer, aka the softmax layer. The last layer is the mapping from feature extractors to category membership.
If you are using 2 cameras or more then you do not need to use machine learning. The dartboard can be modeled as a plane, so you only need two cameras to identify its position. You can simply put a camera above the dartboard, and one on its right. Then you can get the horizontal position from the top camera and the vertical position from the side camera.