Molecular mechanisms by which a fluid closely confined between plane-parallel solid walls tends to order itself in layers parallel with the walls (i.e., stratifies) are investigated by a grand canonical ensemble Monte Carlo method. The walls are composed of individual atoms distributed across each wall according to the (100) plane of a face-centered cubic (fee) crystal. Wall atoms are either rigidly fixed (model A) or thermally coupled (model B) to the film, that is the walls are "soft" on account of intermolecular interactions. As for a film between unstructured (i.e., molecularly smooth) walls [Schoen et nl. J. Chem. Phys, 101, 6865 (1994)] stratification is accompanied by a subtle phase transition manifested as a maximum in density fluctuations at the transition point where packing characteristics of film molecules change in transverse directions. Thus, the transition involves phases with different degrees of transverse fee (100)-like order induced by the walls which act like templates. If the transition involves films comprising only one and two layers, the mechanism of stratification is qualitatively similar to the one previously reported (see above) : Well off the transition point the degree of fee (100)-like order is nearly identical for one- and two-layer films and higher than at the transition point where the film is least ordered. The mechanism of stratification-induced phase transitions is different if it involves thicker films which tend to gain fcc (100)-like order more or less abruptly at the transition point. If wall atoms are not thermally coupled the film may solidify under geometrically favorable conditions. Solidification is not observed in model B under identical thermodynamic conditions.