Throughout quantum mechanics we assume that besides position and momentum, the only other degree of freedom an electron has is its spin--which as you said generates a magnetic dipole field. It possesses no "quadrupole spin" or higher-order degree of freedom.
Let's imagine the electron did have an additional degree of freedom akin to a spin, take the most basic case of a "quadrupole spin" of 1/2, such that it can have a quadrupole moment of "up" or "down" in addition to a dipole moment of "up" or "down." For shorthand let's write "up" and "down" as + and -. (Also assume that this particle is still a fermion--this is actually a nontrivial assumption.) Therefore the electron with normal spin 1/2 and "quadrupole spin" 1/2 would have one of the following spin states: |++>, |+->, |-+>, or |-->. If this were the case, then a hamiltonian that doesn't depend on the magnetic dipole or quadrupole moments would have quadruply degenerate energy levels rather than the usual double. For example, instead of a single S orbital permitting two electrons, it would permit four electrons when we include the quadrupole-1/2 degree of freedom, one each corresponding to |++>, |+->, |-+>, and |-->.
I see, thanks. So can you calculate the magnetic field of a permanent magnet by just summing up little dipoles and integrating over the volume of the magnet? (with the idealization that on average all the dipoles are aligned in the same direction)
Well, if you buy into the approximation that the permanent magnet is basically composed of a bunch of little parallel dipoles, then yes you could do that. Magnetic fields do obey the superposition principle. I'd guess that's a decent approximation for certain kinds of magnets. However, most real magnets behave in ways that would contradict that view. For example, take two ordinary bar magnets and position them side-to-side like this
(Unlike most of the things I talk about on PF, this is actually something you can do and you can't really figure out the real answer in your head). You'll notice that once you push them close enough together, the magnets snap together! This is because they induce quadrupole moments on one another--something that you wouldn't find by superposing a bunch of parallel dipole fields. Also, this would contradict the normal view of a magnet as a solenoid, since oppositely flowing currents repel one another.
One way to understand that experiment is by thinking of the iron (or neodymium, etc.) as composed of many different size regions (grains) each with different nonparallel dipole fields, which can reorient themselves under the influence of an outside magnetic field.