Amplify 1W output to 3W
Yes, the output can be fed to another amplifier. You need to take care of level and impedance matching, and design a two resistor attenuator to do this. Here's what you need to know:


The approximate signal level from the existing output at maximum ouput level (MOL - you can calculate this from the 1W and 8R numbers and verify with scope and a source if you have them available).
The signal input level for your new amp for MOL
The input impedance of your new amp.


Now you calculate the values of the two resistors in your divider such that the lower one of the two is less than 10% of the input resistance and the attenuation reduces the MOL from the existing output to the maximum input for the new amp.

This is the technically correct way of doing it - but sometimes this ends up sounding a bit quiet. This is because many cheap amps operate with a little bit of clipping going on. People don't notice and it makes them sound louder. If you want an easier way, you could probably get away with a 1k trimpot connected as a volume control between the stages, and tweak for a sensible sound level. But it is worth going through the numbers anyway, even if you end up not following them to the last dB.