  A complete census of dusty star-forming galaxies (DSFGs) at early epochs is
necessary to constrain the obscured contribution to the cosmic star formation
rate density (CSFRD), however DSFGs beyond $z \sim 4$ are both rare and hard to
identify from photometric data alone due to degeneracies in submillimeter
photometry with redshift. Here, we present a pilot study obtaining follow-up
Atacama Large Millimeter Array (ALMA) $2\,$mm observations of a complete sample
of 39 $850\,\rm\mu m$-bright dusty galaxies in the SSA22 field. Empirical
modeling suggests $2\,$mm imaging of existing samples of DSFGs selected at
$850\,\rm\mu m - 1\,$mm can quickly and easily isolate the "needle in a
haystack" DSFGs that sit at $z>4$ or beyond. Combining archival submillimeter
imaging with our measured ALMA $2\,$mm photometry ($1\sigma \sim
0.08\,$mJy$\,$beam$^{-1}$ rms), we characterize the galaxies' IR SEDs and use
them to constrain redshifts. With available redshift constraints fit via the
combination of six submillimeter bands, we identify 6/39 high-$z$ candidates
each with $>50\%$ likelihood to sit at $z > 4$, and find a positive correlation
between redshift and $2\,$mm flux density. Specifically, our models suggest the
addition of $2\,$mm to a moderately constrained IR SED will improve the
accuracy of a millimeter-derived redshift from $\Delta z/(1+z) = 0.3$ to
$\Delta z/(1+z) = 0.2$. Our IR SED characterizations provide evidence for
relatively high emissivity spectral indices ($\langle \beta \rangle =
2.4\pm0.3$) in the sample. We measure that especially bright ($S_{850\rm\mu
m}>5.55\,$mJy) DSFGs contribute $\sim10$% to the cosmic-averaged CSFRD from
$2<z<5$, confirming findings from previous work with similar samples.
