The bright visual binary HD 72127 almost certainly lies within or behind the Vela supernova remnant. From 1981 to 1988, we have obtained eight new echelle spectra of HD 72127A near the Ca II K line, along with four similar spectra near the Na I D lines. In addition to the unrivaled intensity and width of this interstellar K line formed at a distance of only d less-than-or-similar-to 500 pc, which were previously discovered by Thackeray, the spectra show clearly the unique temporal variability of the interstellar absorption along this light path, previously reported in Paper I. The new results strengthen further the hypothesis that the interstellar absorption toward this star occurs predominantly in the disturbed gas located within the SNR, in which the interstellar grains have been largely destroyed. The variable interstellar line profiles consist of at least 10 components at the K line and of at least eight, generally corresponding components at the D lines. The total column densities summed over all of these interstellar clouds, N(tot)(Ca II) and N(tot)(Na I), varied irregularly by approximately 45% in 5 years and approximately 38% in 10 years, respectively. Larger variations were seen in the column densities of two of the individual clouds; the largest of these occurred at a low radial velocity, upsilon-LSR almost-equal-to -4 km s-1.