2nd template for posts

I have created a template for a specific post category.

The template lists a brief extract of all posts.

Upon selecting a post the user is shown the complete post and the layout uses single.php.

All works ok, however, I would like to have a 2nd category with a different single post layout. What would the new post template be called if not single.php and how do I get my 2nd category to use it?

Thanks