  We simulate the bond and site percolation models on several three-dimensional
lattices, including the diamond, body-centered cubic, and face-centered cubic
lattices. As on the simple-cubic lattice [Phys. Rev. E, \textbf{87} 052107
(2013)], it is observed that in comparison with dimensionless ratios based on
cluster-size distribution, certain wrapping probabilities exhibit weaker
finite-size corrections and are more sensitive to the deviation from
percolation threshold $p_c$, and thus provide a powerful means for determining
$p_c$. We analyze the numerical data of the wrapping probabilities
simultaneously such that universal parameters are shared by the aforementioned
models, and thus significantly improved estimates of $p_c$ are obtained.
