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// remove the default caption output
add_filter( 'soliloquy_output_caption', 'soliloquy_empty_default_caption', 10, 5 );
function soliloquy_empty_default_caption( $caption, $id, $slide, $data, $i ) {
        return $caption;
}

// add new caption output within the slide item's link tag
add_filter( 'soliloquy_output_after_image', 'soliloquy_caption_inside_slide_link', 10, 5 );
function soliloquy_caption_inside_slide_link( $caption, $id, $slide, $data, $i ) {

  if ( isset( $slide['caption'] ) && !empty( $slide['caption'] ) ) {
		$caption = '<div class="caption">' . $slide['caption'] . '</div>'; 
  } 
  
  return $caption;

}