We study the vortex phase diagram of the single-layer Bi2Sr2CuO6+δ (Bi2201) superconductor by means of bulk magnetization measurements on high-quality oxygen-overdoped crystals. In striking contrast with the results found in the moderately doped two- and three-layer Bi-based cuprates, Bi2201 exhibits a strong temperature-dependent second-peak effect. By means of measurements of the in-plane and out-of-plane first-penetration field we provide direct evidence that this phenomenon is mainly associated to an increase in the electromagnetic anisotropy on warming. The effect of oxygen-doping δ on the vortex phase diagram results in both the irreversibility and second-peak lines shifting to higher temperatures and fields. This enhanced stability of the Bragg glass phase suggests that the interlayer coupling between Cu-O layers increases with δ. In addition, we found that the critical temperature follows the parabolic relation with the number of holes per Cu-O plane that holds for most single- and two-layer cuprates.