The influence of a partial substitution with sulphur into Te sites on the crystal, electronic and magnetic structures of FeTe is investigated by DFT calculations. The results reveal a phase transition from the antiferromagnetic double-stripe order for pure FeTe to the single-stripe order for S-substituted samples, which coincides with the previously observed appearance of the superconducting state. The magnetic transition is caused by the variations of the average chalcogen position in the unit cell. The analyzed normalstate properties of Fe(Te,S) and Fe(Se;S) compounds allow a detection of the well resolved nesting-driven magnetic fluctuations only for superconducting samples, consistent with their antiferromagnetic ground state. Thus, the role of an S-substitution is a suppression of the double-stripe antiferromagnetic order to give rise to the single-stripe correlations, which are associated with an occurrence of superconductivity in Fe(Te,S) solid solutions.