A resonant inelastic x-ray scattering study of overdamped spin excitations in slightly underdoped La 2−x Sr x CuO 4 (LSCO) with x = 0.12 and 0.145 is presented. Three high-symmetry directions have been investigated: (1) the antinodal (0,0) → ( 1 2 ,0), (2) the nodal (0,0) → ( 1 4 , 1 4 ), and (3) the zone-boundary direction ( 1 2 ,0) → ( 1 4 , 1 4 ) connecting these two. The overdamped excitations exhibit strong dispersions along (1) and (3), whereas a much more modest dispersion is found along (2). This is in strong contrast to the undoped compound La 2 CuO 4 (LCO) for which the strongest dispersions are found along (1) and (2). The t − t − t − U Hubbard model used to explain the excitation spectrum of LCO predicts-for constant U/t-that the dispersion along (3) scales with (t /t) 2 . However, the diagonal hopping t extracted on LSCO using single-band models is low (t /t ∼ −0.16) and decreasing with doping. We therefore invoked a two-orbital (d x 2 −y 2 and d z 2 ) model which implies that t is enhanced. This effect acts to enhance the zone-boundary dispersion within the Hubbard model. We thus conclude that hybridization of d x 2 −y 2 and d z 2 states has a significant impact on the zone-boundary dispersion in LSCO.