Spin-liquid behavior was recently suggested experimentally in the moderately one-dimensional organic compound κ-H3(Cat-EDT-TTF)2. This compound can be modeled by the one-band Hubbard model on the anisotropic triangular lattice with t /t 1.5, where t is the minority hopping. It thus becomes important to extend previous studies, that were performed in the range 0 ≤ t /t ≤ 1.2, to find out whether there is a regime where Mott insulating behavior can be found without longrange magnetic order. To this end, we study the above model in the range 1.2 ≤ t /t ≤ 2 using cluster dynamical mean-field theory (CDMFT). We argue that it is important to choose a symmetrypreserving cluster rather than a quasi one-dimensional cluster. We find that, upon increasing t /t beyond t /t ≈ 1.3, the Mott transition at zero-temperature is replaced by a first-order transition separating a metallic state from a collinear magnetic insulating state. Nevertheless, at the physically relevant value t /t 1.5, the transitions toward the magnetic and the Mott insulating phases are very close. The phase diagram obtained in this study can provide a working basis for moderately one-dimensional compounds on the anisotropic triangular lattice.