Short term storage for fresh soup
There is no such thing as "last a little bit longer". You either kill off all bacteria, which makes the soup shelf stable, or you don't, and then you have a 3-5 days of refrigerated lifetime. 

All kinds of canning are only safe when you use recipes which have been tested for safety. I assume your soup recipe has not been tested. What is worse, it is very unlikely that it could ever pass the test, since foods with the right acidity to be hot water canned are rarely eaten as soups - they will taste more like a pickle. 

Your only option seems to be pressure canning. Luckily for you, the USDA has a "make your own soup" pressure canning recipe. You are restricted in the ratio of vegetables and meat to water, but you are allowed to use any vegetable which is approved for canning on its own, and meat is also OK. Herbs and spices are allowed. You can find the actual recipe on the Internet.