The role of local higher-twist τ > 3 spin-1/2 fermionic operators of the strongly coupled N = 4 supersymmetric Yang-Mills theory on the symmetric and antisymmetric deep inelastic scattering structure functions is investigated. The calculations are carried out in terms of the duality between N = 4 SYM theory and type IIB supergravity on AdS₅ × S⁵. Particularly, we explicitly obtain the structure functions for single-trace spin-1/2 fermionic operators in the 20* and 60* irreducible representations of SU(4)R, corresponding to twists 4 and 5, respectively. We also calculate the contributions of other single-trace spin-1/2 fermionic operators in the 4, 20 and 60 irreducible representations of SU(4)R. New important effects are found in comparison with the minimal twist (τ = 3) case, and they are studied thoroughly.