  Phase diagrams and spontaneous magnetization are rigorously calculated for a
coupled spin-electron model on a doubly decorated honeycomb lattice, which
accounts for a quantum-mechanical hopping of the mobile electrons on decorating
sites, the nearest-neighbor Ising coupling between mobile electrons and
localized spins, as well as, the further-neighbor Ising coupling between the
localized spins placed on nodal sites. The spontaneously ordered ferromagnetic
phase, spontaneously ordered antiferromagnetic phase and disordered
paramagnetic phase emerge in a phase diagram depending on an electron filling
of the decorating sites, a relative size of the hopping term and both
considered coupling constants. It is evidenced that a nature and size of the
further-neighbor Ising coupling between the localized spins basically
influences rise and fall of reentrant transitions close to a phase boundary
between the paramagnetic phase and both spontaneously ordered phases.
