The existing definition of generalized intuitionistic fuzzy soft sets (GIFSSs) is clarified and reformulated as a combination of an IFSS over the universe of discourse and an intuitionistic fuzzy set in the parameter set. With this new perspective, two different types of generalized intuitionistic fuzzy soft subsets and various new operations are developed for GIFSSs. The upper and lower substitutions for both intuitionistic fuzzy sets and GIFSSs are defined as well. Some existing notions and results are improved by virtue of these new concepts. Using the expectation score function, two binary relations are proposed for comparing intuitionistic fuzzy values. An algorithm is designed for coping with multiattribute decision making (MADM) problems with a combined use of the GIFSS, the extended intersection operation, the intuitionistic fuzzy weighted averaging operator, and other related notions. A case study concerning a faculty appointment problem is conducted to illustrate the proposed algorithm. Moreover, a comparative analysis between our approach and other related works is given to demonstrate the effectiveness and advantages of the newly proposed method.