How to fix absent Python autocompletion on object instances in Vim?

I have found a strange behavior in Vim when I attempt to use autocompletion on objects. If I instantiate the objects on a module level, the Vim autocompletion will work on the instance I create:

If I try the same from within a function or class, it is no longer working:

Does anyone know how to fix this, or is there a way to get omnicompletion to work on instances in a non module-scope?