Is there a module that can replace tabbed menus with contextual links?
The Custom Contextual Links module does this https://drupal.org/project/ccl 

Jeff Eaton gives a good overview of this module https://www.lullabot.com/blog/article/module-monday-custom-contextual-links

Alternatively, you could theme the existing menu to be styled like contextual links since both use an unordered list, just different css classes that trigger style and javascript behavior.  To do so you could either use a custom module to add the css classes necessary or add your own css rules.