  We present a modified optical luminosity--HI mass bivariate luminosity
function (BLF) based on HI line observations from the Nan\c{c}ay Interstellar
Baryons Legacy Extragalactic Survey (NIBLES), including data from our new, four
times more sensitive follow-up HI line observations obtained with the Arecibo
radio telescope. The follow-up observations were designed to probe the
underlying HI mass distribution of the NIBLES galaxies that were undetected or
marginally detected in HI at the Nan\c{c}ay Radio Telescope. Our total
follow-up sample consists of 234 galaxies, and it spans the entire luminosity
and color range of the parent NIBLES sample of 2600 nearby (900 $< cz <$ 12,000
kms$^{-1}$) SDSS galaxies. We incorporated the follow-up data into the
bivariate analysis by scaling the NIBLES undetected fraction by an Arecibo-only
distribution. We find the resulting increase in low HI mass-to-light ratio
densities to be about 10\% for the bins $-1.0 \le $ log($\frac{M_{\rm
HI}/M_{\odot}}{ L_{\rm r}/L_{\odot}}$) $ \le -0.5$, which produces an increased
HI mass function (HIMF) low mass slope of $\alpha = -1.14 \pm 0.07$, being
slightly shallower than the values of $-1.35 \pm 0.05$ obtained by recent blind
HI surveys. Applying the same correction to the optically corrected bivariate
luminosity function from our previous paper produces a larger density increase
of about 0.5 to 1 dex in the lowest HI mass-to-light ratio bins for a given
luminosity while having a minimal effect on the resulting HIMF low mass slope,
which still agrees with blind survey HIMFs. This indicates that while low
HI-mass-to-light ratio galaxies do not contribute much to the one-dimensional
HIMF, their inclusion has a significant impact on the densities in the
two-dimensional distribution.
