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It is well established that the magnetically structured solar atmosphere supports the propagation of MHD waves along various kind of jets including also the solar wind. It is well-known as well that under some conditions, namely high enough jet speeds, the propagating MHD modes can become unstable against to the most common Kelvin--Helmholtz instability (KHI). In this article, we explore how the propagation and instability characteristics of running along a slow solar wind MHD modes are affected when they are investigated in the framework of the ideal Hall-magnetohydrodynamics. Hall-MHD is applicable if the jet width is shorter than or comparable to the so called Hall parameter $l_\mathrm{Hall} = c/ω_\mathrm{pi}$ (where $c$ is the speed of light and $ω_\mathrm{pi}$ is the ion plasma frequency). We model the solar wind as a moving with velocity $\vec{v}_0$ cylindrical flux tube of radius $a$, containing incompressible plasma with density $ρ_\mathrm{i}$ permeated by a constant magnetic field $\vec{B}_\mathrm{i}$. The surrounding plasma is characterized with its density $ρ_\mathrm{e}$ and magnetic field $\vec{B}_\mathrm{e}$. The dispersion relation of MHD waves is derived in the framework of both standard and Hall-MHD and is numerically solved with input parameters: the density contrast $η= ρ_\mathrm{e}/ρ_\mathrm{i}$, the magnetic fields ratio $b = {B}_\mathrm{e}/{B}_\mathrm{i}$, and the Hall scale parameter $l_\mathrm{Hall}/a$. It is found that the Hall current, at moderate values of $l_\mathrm{Hall}/a$, stimulates the emerging of KHI of the kink ($m = 1)$ and high-mode ($m \geqslant 2$) MHD waves, while for the sausage wave ($m = 0$) the trend is just the opposite---the KHI is suppressed.