The spin bottleneck effect was first observed in vertical double quantum dots with different g factors in high magnetic fields. We further investigate the spin-dependent resonance tunneling through the same quantum dot system in low magnetic fields. There are no resonance tunneling peak lines, even though one of the Zeeman levels is aligned, because the mismatch of the other Zeeman sublevels blocks the resonance tunneling. However, the level broadening effect partially releases the spin-dependent blockade. As a compromise between two effects, one resonance tunneling peak line splits into two peak lines and forms a kink structure. The split of the two current peak lines is half of Zeeman energy difference between two dots.