© 2016, Springer Science+Business Media Dordrecht. We deal with the generalized Orlicz–Morrey space Mϕ,Φ(Rn) of the third kind and consider the decomposition method. Also we characterize its predual space. Some maximal estimates for generalized Orlicz–Morrey spaces of the third kind are also obtained by using the weighted Hardy operators. As an application, we consider the Olsen inequality, which is a bilinear estimate on the fractional integral operator. As an appendix, we consider a general form of the vector-valued boundedness of the Hardy–Littlewood maximal operator, where ϕ in the definition of Mϕ,Φ(Rn) depends on x as well. This paper contains a remedy for the mistake in the proof of the Olsen inequality of the 2014 paper by the second author (Iida et al. in Z. Anal. Anwend. 33(2):149–170, 2014).