I am a newbie, but since no guru got time to answer here, I would like to offer my opinion.
You may need to add more loops to add more categories. Please refer to page 47 of “Smashing WordPress: Beyond the Blog” here at Google Book (https://books.google.com/books?id=_z1UGfWVM20C&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false).
(Title of the topic/chapter is “Three’s a Charm, But Four Loops Are Way Cooler”)
The book got the code, and the author’s conclusion is like this.
“There you have it: four loops starting with one for featured posts, and then three more to display the latest from three different categories. This code can easily be implemented in just about any
WordPress theme, but you would need to alter it to fit your category setup to make it work properly.”
The example code gives you only 4 categories, but you can just add more. The method obvious requires coding.