The most extensive survey of carbon monoxide (CO) gas in the Taurus molecular cloud relied on 12CO and 13CO J = 1 → 0 emission only, distinguishing the region where 12CO is detected without 13CO (named mask 1 region) from the one where both are detected (mask 2 region) (Goldsmith et al. 2008; Pineda et al. 2010). We have taken advantage of recent 12CO J = 3 → 2 James Clerk Maxwell Telescope observations, where they include mask 1 regions to estimate density, temperature, and N(CO) with a large velocity gradient model. This represents 1395 pixels out of ∼1.2 million in the mark 1 region. Compared to Pineda et al. (2010) results and assuming a Tkin of 30 K, we find a higher volume density of molecular hydrogen of 3.3 × 103 cm−3, compared to their 250–700 cm−3, and a CO column density of 5.7 × 1015 cm−2, about a quarter of their value. The differences are important and show the necessity to observe several CO transitions to better describe the intermediate region between the dense cloud and the diffuse atomic medium. Future observations to extend the 12CO J= 3 → 2 mapping further away from the 13CO-detected region comprising mask 1 are needed to revisit our understanding of the diffuse portions of dark clouds.