  This paper is concerned with the asymptotics of resonances in the
semiclassical limit $h\to 0^+$ for $2\times 2$ matrix Schr\"odinger operators
in one dimension. We study the case where the two underlying classical
Hamiltonian trajectories cross tangentially in the phase space. In the setting
that one of the classical trajectories is a simple closed curve whereas the
other one is non-trapping, we show that the imaginary part of the resonances is
of order $h^{(m_0+3)/(m_0+1)}$, where $m_0$ is the maximal contact order of the
crossings. This principal order comes from the subprincipal terms of the
transfer matrix at crossing points which describe the propagation of microlocal
solutions from one trajectory to the other. In addition, we compute explicitly
the leading coefficient of the resonance widths in terms of the probability
amplitudes associated with all the \textit{generalized classical trajectories}
escaping to infinity from the closed trajectory.
