  We study Smith-Purcell radiation from a conducting grating generated by a
vortex electron with an orbital angular momentum $\ell \hbar$, described as a
generalized Laguerre-Gaussian packet, which has an intrinsic magnetic dipole
moment and an electric quadrupole moment. By using a multipole expansion of the
electromagnetic field of such an electron, we employ a generalized
surface-current method, applicable for a wide range of parameters. The radiated
energy contains contributions from the charge, from the magnetic moment, and
from the electric quadrupole moment, as well as from their interference. The
quadrupole contribution grows as the packet spreads while propagating, and it
is enhanced for large $\ell$. In contrast to the linear growth of the radiation
intensity from the charge with a number of strips $N$, the quadrupole
contribution reveals an $N^3$ dependence, which puts a limit on the maximal
grating length for which the radiation losses stay small. We study
spectral-angular distributions of the Smith-Purcell radiation both analytically
and numerically and demonstrate that the electron's vorticity can give rise to
detectable effects for non-relativistic and moderately relativistic electrons.
On a practical side, preparing the incoming electron's state in a form of a
non-Gaussian packet with a quadrupole moment -- such as the vortex electron, an
Airy beam, a Schr\"odinger cat state, and so on -- one can achieve quantum
enhancement of the radiation power compared to the classical linear regime.
Such an enhancement would be a hallmark of a previously unexplored quantum
regime of radiation, in which non-Gaussianity of the packet influences the
radiation properties much stronger than the quantum recoil.
