  We have complemented existing observations of HI absorption with new
observations of HCO$^+$, C$_2$H, HCN, and HNC absorption from the Atacama Large
Millimeter/submillimeter Array (ALMA) and the Northern Extended Millimeter
Array (NOEMA) in the direction of 20 background radio continuum sources with
$4^\circ \leq |b| \leq 81^\circ$ to constrain the atomic gas conditions that
are suitable for the formation of diffuse molecular gas. We find that these
molecular species form along sightlines where $A_V \gtrsim 0.25$, consistent
with the threshold for the HI-to-H$_2$ transition at solar metallicity.
Moreover, we find that molecular gas is associated only with structures that
have an HI optical depth $> 0.1$, a spin temperature $< 80$ K, and a turbulent
Mach number $\gtrsim 2$. We also identify a broad, faint component to the
HCO$^+$ absorption in a majority of sightlines. Compared to the velocities
where strong, narrow HCO$^+$ absorption is observed, the HI at these velocities
has a lower cold neutral medium (CNM) fraction and negligible CO emission. The
relative column densities and linewidths of the different molecular species
observed here are similar to those observed in previous experiments over a
range of Galactic latitudes, suggesting that gas in the solar neighborhood and
gas in the Galactic plane are chemically similar. For a select sample of
previously-observed sightlines, we show that the absorption line profiles of
HCO$^+$, HCN, HNC, and C$_2$H are stable over periods of $\sim 3$ years and
$\sim 25$ years, likely indicating that molecular gas structures in these
directions are at least $\gtrsim 100$ AU in size
