  ALMA observations of the Sun at mm-$\lambda$ offer a unique opportunity to
investigate the temperature structure of the solar chromosphere. In this
article we expand our previous work on modeling the chromospheric temperature
of the quiet Sun, by including measurements of the brightness temperature in
the network and cell interiors, from high resolution ALMA images at 3 mm (Band
3) and 1.26 mm (Band 6). We also examine the absolute calibration of ALMA
full-disk images. We suggest that the brightness temperature at the center of
the solar disk in Band 6 is $\sim440$ K above the value recommended by White et
al. (2017) and we give improved results for the electron temperature variation
of the average quiet Sun with optical depth, as well as the derived spectrum at
the center of the disk. We found that the electron temperature in the network
is considerably lower than predicted by model F of Fontenla et al. (1993) and
that of the cell interior considerably higher than predicted by model A.
Depending upon the network/cell segregation scheme, the electron temperature
difference between network and cell at $\tau=1$ (100 GHz) is from $\sim$660 to
$\sim$1550 K, compared to $\sim$3280 K predicted by the models; similarly, the
$T_e$ ratio is from $\sim$1.10, to 1.24, against $\sim$1.55 of the model
prediction. We also found that the network/cell $T_e(\tau)$ curves diverge as
$\tau$ decreases, indicating an increase of contrast with height and possibly a
steeper temperature rise in the network than in the cell interior.
