  We consider solutions of the matrix KP hierarchy that are trigonometric
functions of the first hierarchical time $t_1=x$ and establish the
correspondence with the spin generalization of the trigonometric Calogero-Moser
system on the level of hierarchies. Namely, the evolution of poles $x_i$ and
matrix residues at the poles $a_i^{\alpha}b_i^{\beta}$ of the solutions with
respect to the $k$-th hierarchical time of the matrix KP hierarchy is shown to
be given by the Hamiltonian flow with the Hamiltonian which is a linear
combination of the first $k$ higher Hamiltonians of the spin trigonometric
Calogero-Moser system with coordinates $x_i$ and with spin degrees of freedom
$a_i^{\alpha}, \, b_i^{\beta}$. By considering evolution of poles according to
the discrete time matrix KP hierarchy we also introduce the integrable discrete
time version of the trigonometric spin Calogero-Moser system.
