A cheap 2x teleconverter(not a lens) will degrade some of the image quality and slow down the focusing. The worst part will be with getting a sharp photo due to extended range. Your S3IS is already 432mm(35mm equivalent) and that's quite a distance there. Your cameras image stabilization can only do so much and is probably pushing it with that much reach.
If you add a 2x teleconverter you will run into problem with image quality deterioration and somewhat blurry photos due to camera shake or the much higher needed shutter speed needed. I'm betting this will also take off 2 stops of light, just like a 2x teleconverter for a DSLR, which means that it will cut your shutter speed down to one fourth of the speed it would have been for a proper exposure if you didn't have the teleconverter on. This will cause more problems for getting a fast enough shutter speed. The only time you could get a decently sharp pic with that on would be on a really bright day and that's it.
It's not worth the headache for either one. People pay over $6000 to get lens for their DSLR that is good enough for a sharp photo without even attaining the reach you are wanting your point and shoot to do.
To give you an example, here is the Canon 500mm F4 IS lens(which on a lower end canon has a 35mm equivalent to 800mm). A 2x teleconverter on your S3IS would make it up to a 35mm equivalent of 864mm. This Canon 500mm F4 also is a 1.5 feet long and weighs 8 1/2 pounds. It's a behemoth.
This telephoto lens:
has about the power of this lens (first one):
The obvious physical and price differences must mean the cheaper one will not do what the more expensive one will do. What is probably lacking in the first lens?
Thanks very much for any enlightenment for this beginner, I'm using a Canon S3 IS.