Sure. Cut up the old mold, some people use a meat grinder, and use the bits to fill a new batch. Make a thin glove coat of straight silicone first, then back it up with the filled batch.
Just make sure you use the same type of silicone you cut up. A different durometer say a heavier silicone will sink and touch the part causing voids and air bubbles
as Robin mentioned, it is very important to grind them up. In the past I have (in an attempt to save time) layed in larger pieces. this works sometimes and has the advantage that you don't get any bubbles, but very often the old silicone may have had oils from resin or traces of talcum or other removal agents, which does not allow the new silicone to meld to the old. If the old silicone is completly encased this may not have any side effects, but if it is near the surface chances are that the two silicones will split apart from each other.
Grinding the old silicone does tend to create more air pockets, but if you coat your object first this should not create any issues.
Also as mentioned by Bounty Hunter, DON'T mix different types of silicone. I did this with normal silicone and non-fluid silicone... needless to say they did not stick to one another at all... a lot of work for nothing and back to the drawing board.