The intra-atomic magnetic dipole moment - frequently called ⟨Tz⟩ term - plays an important role in the determination of spin magnetic moments by x-ray absorption spectroscopy for systems with nonspherical spin density distributions. In this work, we present the dipole moment as a sensitive monitor to changes in the electronic structure in the vicinity of a phase transiton. In particular, we studied the dipole moment at the Fe(2+) and Fe(3+) sites of magnetite as an indicator for the Verwey transition by a combination of x-ray magnetic circular dichroism and density functional theory. Our experimental results prove that there exists a local change in the electronic structure at temperatures above the Verwey transition correlated to the known spin reorientation. Furthermore, it is shown that measurement of the dipole moment is a powerful tool to observe this transition in small magnetite nanoparticles for which it is usually screened by blocking effects in classical magnetometry.