Amazon Kindle Fire - Control Music playback without unlocking

From your homescreen, press Music – Menu – Settings. Then press Lock screen controls. Ensure that a tick is in the box. Then go back to your music library and play a song.

To test it, press the Lock button on the side, then press it again to Unlock. You will need to move the slider once and this will display the music controls.

To unlock the screen properly, press Unlock at the top right of the screen.