I think the best way would be to double click the component several times until you have selected the faces you want to paint.
With this you have been able to edit the component.
That is the context you want to apply the new material, applying it to the faces inside the component instead painting the component.
The difference is that when you paint a component, only the faces that inside the component AND have no material assigned, receive the new material. All faces inside the component that have a material directly assigned, will ignore the component's material as box described.