Let \(\text{Map}_{\mathcal T}(K,X)\) denote the mapping space of continuous based functions between two based spaces \(K\) and \(X\), where \(K\) is assumed a finite complex. Recently, G. Arone [Trans. Am. Soc. 351, 1123-1250 (2002; Zbl 0945.55011)] has given an explicit model for the Goodwillie tower of the functor sending a space \(X\) to the suspension spectrum \(\Sigma^{\infty}\text{Map}_{\mathcal T}(K,X)\). Applying a generalized homology theory \(h_{*}\) or similarly a generalized cohomology theory to the Goodwillie tower yields a spectral sequence, which under suitable conditions converges strongly to \(h_{*}\text{Map}_{\mathcal T}(K,X)\). In the present paper, the authors study how important natural constructions on mapping spaces, like product constructions and evaluation map constructions, induce extra structures on the Goodwillie towers and in turn lead to extra structures in the associated spectral sequences. The paper includes results from the first author’s Ph. D. thesis, University of Virginia (2000).