We present a full phase diagram for the one-dimensional (1D) to three-dimensional (3D) crossover of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in an attractive Hubbard model of 3D-coupled chains in a har- monic trap. We employ real-space dynamical mean-field theory which describes full local quantum fluctuations beyond the usual mean-field and local density approximation. We find strong dimensionality effects on the shell structure undergoing a crossover between distinctive quasi-1D and quasi-3D regimes. We predict an optimal regime for the FFLO state that is considerably extended to intermediate interchain couplings and polarizations, directly realizable with ultracold atomic gases. We find that the 1D-like FFLO feature is vulnerable to thermal fluctuations, while the FFLO state of mixed 1D-3D character can be stabilized at a higher temperature.