The Frobenius complexity of a local ring R measures asymptotically the abundance of Frobenius actions of order e on the injective hull of the residue field of R. It is known that, for Stanley-Reisner rings, the Frobenius complexity is either -infinity or 0. However, a complete description of the Frobenius complexity sequence c_ e(R)  for all values of e is not yet known when R is Stanley-Reisner. We will provide the answer for a large class of Stanley-Reisner rings, generalizing work of Alvarez Montaner, Boix and Zarzuela.