  We present the Green Bank Telescope absorption survey of cold atomic hydrogen
($\lesssim 300$K) in the inner halo of low-redshift galaxies. The survey aims
to characterize the cold gas distribution and to address where condensation -
the process where ionized gas accreted by galaxies condenses into cold gas
within the disks of galaxies - occurs. Our sample consists of 16 galaxy-quasar
pairs with impact parameters of $\le$ 20kpc. We detected an HI absorber
associated with J0958+3222 (NGC 3067) and HI emission from six galaxies. We
also found two \ion{Ca}{2} absorption system in the archival SDSS data
associated with galaxies J0958+3222 and J1228+3706, although the sample was not
selected based on the presence of metals in absorption. Our detection rate of
HI absorbers with optical depths of $\ge 0.06$ is $\sim$7\%. We also find that
cold HI phase ($\lesssim$300K) is 44($\pm$18)\% of the total atomic gas in the
sightline probing J0958+3222. We find no correlation between the peak optical
depth and impact parameter or stellar and HI radii normalized impact
parameters, $\rho/\rm R_{90}$ and $\rho/\rm R_{HI}$. We conclude that the
process of condensation of inflowing gas into cold ($\lesssim$ 300K) HI occurs
at the $\rho << 20$kpc. However, the warmer phase of neutral gas (T $\sim$
1000K) can exists out to much larger distances as seen in emission maps.
Therefore, the process of condensation of warm to cold HI is likely occurring
in stages from ionized to warm HI in the inner halo and then to cold HI very
close to the galaxy disk.
