We report a numerical investigation of how the dispersion slope affects the soliton spectral tunneling (SST) in a photonic crystal fiber with three zero dispersion wavelengths. It is discovered that a larger dispersion slope makes group-velocity mismatch between the initial soliton and the transferred wave thereby suppressing the SST effect, while a proper decrease of the dispersion slope enhances the SST effect to widen a supercontinuum range. Besides, we find a soliton-like leaking dispersion wave, which can sustain information and energy for a short time within a particular spectral range.