The first problem is the path, you need override the component from: **your_site/components/com_contact**
not **your_site/administrator/components/com_contact**

Also the code for the template must be in: **your_site/templates/your_template**

Here you have an example: <br/>http://stackoverflow.com/questions/20586581/adding-fields-to-joomla-3-x-contact-component

Other possible solution is via plugin, find more info here:
https://docs.joomla.org/Adding_custom_fields_to_core_components_using_a_plugin