  A dozen X-ray supernova shock breakout (SN SBO) candidates were reported
recently based on XMM-Newton archival data, which increased the X-ray selected
SN SBO sample by an order of magnitude. Assuming they are genuine SN SBOs, we
study the luminosity function (LF) by improving upon the method used in our
previous work. The light curves and the spectra of the candidates were used to
derive the maximum volume within which these objects could be detected with
XMM-Newton by simulation. The results show that the SN SBO LF can be described
by either a broken power law (BPL) with indices (at the 68$\%$ confidence
level) of $0.48 \pm 0.28$ and $2.11 \pm 1.27$ before and after the break
luminosity at $\log (L_b/\rm erg\,s^{-1})=$ $45.32 \pm 0.55$ or a single power
law (SPL) with index of $0.80 \pm 0.16$. The local event rate densities of SN
SBOs above $5\times 10^{42}$ $\rm erg\,s^{-1}$ are consistent for two models,
i.e., $4.6 ^{+1.7}_{-1.3} \times 10^4$ and $4.9 ^{+1.9}_{-1.4} \times 10^4$
$\rm Gpc^{-3}\,yr^{-1}$ for BPL and SPL models, respectively. The number of
fast X-ray transients of SN SBO origin can be significantly increased by the
wide-field X-ray telescopes such as the Einstein Probe.
