How can I connect a new ground rod?
In reading your question, I think we are into that area of all the things we call "grounding" which are actually bonds, jumpers and grounded equipment. What you need to concentrate on is NEC Part III Grounding Electrode System. 

I don't expect a DIY to know the entire section, but the highlights goes as follows. The entire GES needs to terminate at the first point of disconnect in your system. If you open up your panel you probably see a large #8 or #6 Bare copper conductor (the Grounding Electrode Conductor) attached to your neutral bus. This is where your cold water ground should be attached. You need to attache your driven ground rod to the bare conductor. 

The driven rod you are proposing is known as a supplemental or secondary ground to the cold water ground so you do not need to install two or more ground rods. It may become the main ground if you water pipe begins to get cut up and replaced by non metallic water lines. The NEC recommends that you should use as many methods of grounding as you can to any system.

In order to help you visualize what I just said I have attached a generic drawing of a GES. Just ignore the Lighting Protection part.
 

Hope this helps and good luck.