CW-cavity ring down spectroscopy was used to record in a free jet expansion the spectrum of the absorption band in ($^{12}$C$_2$H$_2$)$_2$ with origin at 6547.6 cm$^{-1}$. It is a perpendicular band and corresponds to 2CH excitation in the hat unit of the T-shaped dimer. Calibration (better than $±1\times 10^{-3}$ cm$^{-1}$ accuracy) and ring-down time (130 $μ$s) were improved compared to a previous contribution , \textit{Mol. Phys.} 2010, \textbf{108}, 2158-2164}, and a line-by-line analysis was undertaken. Three series of lines were identified involving levels with A1$^+$, E$^+$ and B1$^+$ tunneling symmetries, confirming the spectral and symmetry analyses reported in the literature for the 1CH excitation band , \textit{J. Chem. Phys.} 1988, \textbf{89}, 6028-6045}. 164 vibration-rotation-tunneling lines were assigned in the $ K_a'-K_a''= 2-3, 0-1, 2-1$ and $4-3$ sub-bands and effective rigid rotor vibration-rotation constants were obtained by simultaneously fitting 1CH and 2CH lines from the same symmetry series. Perturbations affecting the $K_a$ stacks, in particular, are reported. The tunneling frequency in 2CH is estimated to be $ν^{2CH}_{tunn.}$ = 270 MHz for the $K_a$ = 0 stack. The rotational temperature is determined to be 23 K from relative line intensities and the half lifetime of the dimer in the 2CH hat state is estimated to 1 ns from individual line widths. ; Author Institution: Service de Chimie quantique et Photophysique CP160/09, Faculte des Sciences, Universite Libre de Bruxelles (U.L.B.), Av. Roosevelt, 50, B-1050, Bruxelles, Belgium; ACTMM-CC 15.01, Institut Charles Gerhardt, 34095 Montpellier, France