  A recent segregation criterion [V. Garz\'o, Phys. Rev. E \textbf{78},
020301(R) (2008)] based on the thermal diffusion factor $\Lambda$ of an
intruder in a heated granular gas described by the inelastic Enskog equation is
revisited. The sign of $\Lambda$ provides a criterion for the transition
between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE).
The present theory incorporates two extra ingredients not accounted for by the
previous theoretical attempt. First, the theory is based upon the second Sonine
approximation to the transport coefficients of the mass flux of intruder.
Second, the dependence of the temperature ratio (intruder temperature over that
of the host granular gas) on the solid volume fraction is taken into account in
the first and second Sonine approximations. In order to check the accuracy of
the Sonine approximation considered, the Enskog equation is also numerically
solved by means of the direct simulation Monte Carlo (DSMC) method to get the
kinetic diffusion coefficient $D_0$. The comparison between theory and
simulation shows that the second Sonine approximation to $D_0$ yields an
improvement over the first Sonine approximation when the intruder is lighter
than the gas particles in the range of large inelasticity. With respect to the
form of the phase diagrams for the BNE/RBNE transition, the kinetic theory
results for the factor $\Lambda$ indicate that while the form of these diagrams
depends sensitively on the order of the Sonine approximation considered when
gravity is absent, no significant differences between both Sonine solutions
appear in the opposite limit (gravity dominates the thermal gradient). In the
former case (no gravity), the first Sonine approximation overestimates both the
RBNE region and the influence of dissipation on thermal diffusion segregation.
