I did research and apparently some cameras automatically rewind the film which saves the already taken photos.
Taking a quick look at the Canon SureShot 60 Zoom Manual it appears that your camera does not "preload" all of the film onto the camera's takeup spool and then wind it back into the film cartridge as you shoot. Instead, the SureShot 60Zoom only winds the film out of the cartridge as it advances the film frame by frame after you take each photo.
It can sense when the end of a roll is reached by the resistance from the end of the film being taped to the spindle inside the film cartridge. It will then automatically rewind the film back into the cartridge. Since you opened the camera's back before the end of the roll was reached, the camera never rewound anything back into the cartridge.
What most likely happened when you closed the back is that the camera assumed you had just loaded a fresh roll of film and advanced the film enough to move the film that was already outside the cartridge when you closed the back past the film gate (the hole where the light from the lens comes through when the shutter is opened). With that particular camera you could have started taking additional photos from there and they would probably have been fine.
By advancing the film another 5-6 frames, you just used more unexposed film. The remaining film should be usable, though I would not use it to take any photos of "irreplaceable" events.
The images that you took before you opened the back of the camera are the real question mark. The most recent two images are almost certainly totally gone. Depending upon how tightly the used film was wound on the camera's takeup spool, how long the camera was open, and how bright the light you were in was at the time: the earlier images may be salvageable, though they will probably show some evidence of light leaks that will have affected parts of those frames.
