Poly(N-isopropylacrylamide) (PNIPAM) is well known to exhibit reentrant behavior or cononsolvency in response to the composition of a mixed solvent consisting of water and a low-chain alcohol. Since the solvent structure plays an important role in this phenomenon, the presence of structure-breaking/structure-making ions in solution is expected to have a dramatic effect on the cononsolvency of PNIPAM. The present work examines the way that the presence of different salts can modify the reentrant-phase diagram displayed by a cationic PNIPAM microgel in the mixed ethanol/water solvent. The effects of four Hofmeister anions--SO(4)(2-), Cl(-), NO(3)(-) and SCN(-)--with different abilities to modify the solvent structure are analyzed. The species with kosmotropic or structure-making character show a clear competition with ethanol for the water molecules, intensifying the nonsolvency of the PNIPAM with the EtOH volume fraction (phi(e)). However, striking results are found with the most chaotropic or structure-breaking anion, SCN(-). In contrast to what happens in water-rich solutions, the presence of SCN(-) in alcohol-rich solvents enhances the solubility of the polymer, which macroscopically results in the microgel swelling. Moreover, this ion displays great stabilizing properties when phi(e)> is 0.2. These results have been explained by considering how chaotropic or structure-breaking ions interact with water and ethanol molecules.