A wee little counter point I might add here. I used a Sony H5 (basically identical to your S3) for 3 years prior to my DSLR, and I also extensively used a Sony HGD1758 1.7x teleextender with it for wildlife shooting and random vacation stuff requiring the big zoom reach. If you get a high quality add-on lens, you can get pretty nice results - even for enthusiasts who know a bit about cameras and image quality. I agree about avoiding the cheap models, but the better add-ons like Sony, Raynox, and Olympus I would definitely recommend (in case you didn't realize, you can actually use any of those brands on your S3 - you DON'T have to buy a Canon add-on - as long as they have the same 58mm thread size as your extension tube, you can use other brands...but you must have the Canon extension tube).
As for the newer cameras with the 'bigger zoom'...be wary. First off, many of these newer cameras have not been faring quite as well in reviews and with many of the enthusiasts in image quality. The excessive race to cram ever-more pixels on tiny little sensors is resulting often in worse image quality for fine details - excessive noise, very aggressive noise reduction which eats away details, and some artifacting from blooming on the small pixel sites, with often increased purple fringing and other chromic abberations. I wouldn't say ALWAYS, but certainly some of the newer models have not compared quite as favorably as their previous ancestors. Your S3 was considered by many to be at or near the high point for the SxIS line.
Also, don't just go by the multiplication factor of the lens to determine actual telephoto range. A 15x lens is NOT always longer at the telephoto lens compared to a 12x lens. That multiplication factor must first take into consideration what the native wide end of the lens is. Many of the newer 15x-20x lenses start much wider than your camera (which can be a wonderful advantage at the wide end, while possibly not reaching as far at the longer end). Where your S3 starts at 36mm on the wide end...a 12x zoom gives it 432mm at the tele end. A new camera may start at 27mm for example, and carry a 15x zoom. Multiply those factors and you get 405mm...LESS than your S3 at max telephoto. Some of the 20x cameras that start at 27mm or 28mm may actually reach farther than your S3, getting out to 540-560mm. However, if you were to add a 1.7x teleextender to your S3, at full zoom, you'd actually be reaching out to 720mm!
Just keep all of that in mind when shopping. If you want some ideas of how a superzoom with the same lens range as your S3 can perform with a decent extender lens, I have a wildlife/bird gallery that was shot with my Sony H5 and in almost every case, the Sony HDG1758 teleextender (there may be a few shots in there without it):
http://www.pbase.com/zackiedawg/h5wakodahatchee&page=all
Decide for yourself if you think you'd rather hold onto your current camera and add the extender lens, or invest in a newer cam with a bigger lens range...it's up to you!