Same menu display only a couple of levels down when used normally but when used within the "Site map" module it would display all the levels?
You can install the Menu Block module to help with that.

You can define a define a block for a specific menu, and in the block configuration you can choose the maximum depth (see the active select list in the image below). Then you can add that block to your header/whatever region in place of the menu that's there now.



This won't affect the underlying menu itself though, just the display of it for that particular block. So in the sitemap, all levels of links will show. 

Obviously you may need to make a few CSS tweaks to your menu as the wrapping classes will probably have changed, but I think this is by far the quickest method to get what you're looking for.