How handle the add of a new feature to the dataset? Let's say that you trained a model (eg. Random Forest) on a dataset with ten features (or columns).  

Now you add one or more features to the dataset. You need the information brought from these new features so you want to enrich your model.  

What happens to the data collected till now? How can I use it? Those data have just ten features. How can I use them to train the new model?
What you ask for is known as Transfer Learning in the Machine Learning framework, so you might want to look more into that direction.

An interesting publication regarding Transfer Learning in Decision Trees is this.