Fiber-based wavelength converters employing soliton self-frequency shift (SSFS) are particularly attractive as tunable mid-infrared (MIR) sources due to their compact design and robust nature. Early attempts of wavelength conversion utilizing SSFS in the MIR spectral region involved cascading silica and fluoride nonlinear fibers (NLFs). In this work, we take a step forward and demonstrate SSFS in a cascade involving three types of NLFs made of three different glasses, that is, silica, fluoride, and chalcogenide (ChG). Due to its high nonlinearity, incorporating ChG NLF in a cascade significantly reduces the pump pulse energy required for achieving a target SSFS, thereby greatly simplifying the wavelength converter. The dispersion and nonlinearity of the NLFs convert a fundamental soliton in one NLF to become a high order soliton in the subsequent NLF, allowing soliton fission followed by SSFS at each stage of the cascade. Stepwise SSFS in the cascade eliminates the need for amplification between NLFs, further simplifying the wavelength converter. In the first stage, a pump pulse with an energy of 10.6 nJ at a wavelength of 1.94 mu m initiates SSFS in a silica NLF, achieving tunable solitons up to a wavelength of 2.14 mu m with a wavelength offset of 200 nm (14.5 THz). In the second stage, a ZBLAN NLF extends the SSFS up to a wavelength of 2.52 mu m, adding a wavelength offset by 380 nm (21.1 THz). In the third stage, a ChG NLF allows further SSFS up to a wavelength of 3.28 mu m, further increasing the wavelength offset by 760 nm (27.6 THz). This work is an important step for the development of fiber-based compact soliton sources in the MIR.