Hi Bernward,
I have checked and found that flymenu it loaded before the single post loop. So it is not possible to get content via gd_output_location shortcode there.
You can try following steps to achieve it.
1) Add Text widget in Widgets > Main Sidebar with following content:
<div id="_gd_snippet_move_to_flymenu">[gd_output_location location="detail"]</div>
<script type="text/javascript">if(jQuery('#_gd_snippet_move_from_flymenu').length && jQuery('#_gd_snippet_move_to_flymenu').length){jQuery('#_gd_snippet_move_from_flymenu').html(jQuery('#_gd_snippet_move_to_flymenu').html());jQuery('#_gd_snippet_move_to_flymenu').closest('.widget_text').remove();}</script>
2) Add following content in fly menu content instead of gd_output_location shortcode.
<div id="_gd_snippet_move_from_flymenu"> </div>
Regards,
Kiran