In the quest for the construction of silicene, the silicon analogue of graphene, recent experimental studies have identified a number of distinct ultrathin Si over-layer structures on a Ag(111) surface. Here we use first-principles calculations to probe associated atomic-scale mechanisms that can give rise to this rich behavior of Si wetting layers. We find that the interaction between the Si film and the Ag substrate, neither too strong nor too weak, combined with the possibility of buckling, allows for the incorporation of a number of excess Si adatoms in continuous overlayers with a honeycomb network topology. Depending on the Si coverage, we thus obtain a hierarchy of Si mono-atomic films, in agreement with experiments.