In short, no - there isn't a way to get any better photos out of that camera since you aren't allowed to use flash in gymnastics.
As others have mentioned, there is currently no good low light sports digicam on the market - period. The Fuji F30 is the only digicam that has decent (note I say decent) ISO 800 and 1600 performance. It has ISO 3200 but that's pretty much a last ditch once in a while solution - i.e. it works great with extreme close-ups of faces and such where digital noise is not prevalant but will work less well in a situation like yours where there are dark backgrounds and mats and shadows where noise becomes the most prevalant. And, 108mm means you would have to be on the floor right near the aperatus to get usable action shots - I'm guessing by your first post you don't have that kind of access.
Now, let's get to some more specifics. When I look at the exif I see an aperture of around 2.7 and a shutter speed of 1/60 and ISO of 800 - that's just too slow. You need at least 1/250 to 1/320 and even then you'll get blurred hands and feet.Now, to get speeds of 1/250 - you need 2 full stops better. That means you would either need an ISO of 3200 (3200 is two stops better than 800) or an aperture of 1.4 (1.4 is two stops better than 2.8).
And, even then the photos are going to be marginal because 1/250 still isn't good enough to stop full motion.
There is no digicam I'm aware of that has aperture of 1.4. And the only digicam that has ISO 3200 that isn't complete trash is the F30 - and as already mentioned it's too short unless you're right by the aperatus. There is another camera from Fuji coming out in the fall that is supposed to combine superzoom capability with the F30s high ISO capability - but it isn't out yet. And even when it is, gymnastics at the youth level is probably one of the worst sports to photograph because of the poor light.
So, I know this sounds bad. What can you do? You can pick your shots - put the camera down to ISO 400 and select points in your son's routine where there isn't a lot of movement - high peaks in his horse work, stops in motion on the rings or after a tumble on the mat. That's the best you can do with this sport.
Don't feel bad though - I have one of the best high ISO cameras on the market today - the Canon 20D. I shoot ISO 1600 or 3200 at gymnastics with $900+ lenses and I get mediocre results. The combination of terrible light and in most cases distance from the events prevents less expensive solutions.
And, by the way - you have an excellent camera. You just have a need to photograph possibly one of the most difficult subjects. It's not a knock on the S3 that it isn't up to such a daunting task.