  High-resolution observations of the solar chromosphere at millimeter
wavelengths are now possible with the Atacama Large Millimeter Array (ALMA),
promising to tackle many open problems in solar physics. Observations from
other ground and space-based telescopes will greatly benefit from coordinated
endeavors with ALMA, yet the diagnostic potential of combined optical,
ultraviolet and mm observations has remained mostly unassessed. In this paper
we investigate whether mm-wavelengths could aid current inversion schemes to
retrieve a more accurate representation of the temperature structure of the
solar atmosphere. We performed several non-LTE inversion experiments of the
emergent spectra from a snapshot of 3D radiation-MHD simulation. We included
common line diagnostics such as CaII H, K, 8542 \AA$\,$ and MgII h and k,
taking into account partial frequency redistribution effects, along with the
continuum around 1.2 mm and 3 mm. We found that including the mm-continuum in
inversions allows a more accurate inference of temperature as function of
optical depth. The addition of ALMA bands to other diagnostics should improve
the accuracy of the inferred chromospheric temperatures between $\log
\tau\sim[-6,-4.5]$ where the CaII and MgII lines are weakly coupled to the
local conditions. However, we found that simultaneous multi-atom, non-LTE
inversions of optical and UV lines present equally strong constraints in the
lower chromosphere and thus are not greatly improved by the 1.2 mm band.
Nonetheless, the 3 mm band is still needed to better constrain the mid-upper
chromosphere.
