I don't think you can say much other than showing existence and uniqueness. You can check whether two charts belong to the same maximal smooth atlas. But I don't know what you mean by finding it or showing what it is.
I am having trouble with some definitional stuff, I am using Lee's Introduction to Smooth Manifolds if that helps. I know how to check if such a structure exists, but I don't know how to show what it is.
Thanks