Within the framework of the perturbation expansion of the linear-chain Hubbard model to fourth order in t/U, we consider the effect of second-neighbor interactions \ensuremath{\propto}(t/U
)4 on the spin-Peierls transition temperature
Tc in spin-1/2 Heisenberg chains. We find that the results can be interpreted in terms of a reduced ``effective'' nearest-neighbor antiferromagnetic exchange interaction when the spin-spin interactions are treated in the Hartree-Fock approximation. In the boson algebra approach, we show that this reduced interaction is of a different functional form than in the Hartree-Fock approach and is manifested in a decreased spin-wave velocity c. For typical couplings of the spins to the lattice, both approaches yield a spin-phonon interaction constant greater than the nearest-neighbor value of \ensuremath{\lambda}, but again with different functional forms and the Hartree-Fock
Tc is raised by the second-neighbor interactions, by as much as 30% for \ensuremath{\lambda}=0.5 and (t/U
)2\ensuremath{\simeq}0.1. This is discussed in the light of the fact that, in the Cross and Fisher approach, we find
Tc to be independent of
J2.