salt as sodium chloride does not affect the "hardness" of water as measured by your standard aquarium test kits.
there are two types of 'hardness': GH and KH. GH is a measure of the calcium and magnesium in your water and KH is a measure of the carbonate level. unfortunately, while the test kit you're using measures only the presence of Ca and Mg, there may be many more cationic species in your water than you're able to test for. some of these may include: Sodium, Potassium, Titanium, Nickel, Aluminium, Manganese, Copper, Tin.
the sum total of all of these cations represent your total dissolved solids (TDS) and it is THIS function that has the most significant impact on fish ... much more so than Ca and Mg alone. these cations are attached to anion species such as carbonate but may also be silicate, phosphate, sulphate, chloride and others. again, while you're measuring carbonate as KH there are other disolved inorganic compounds in your water that you're not testing for.
when you run water through a home water softener, the anions Ca++ and Mg++ are exchanged for sodium (Na+) but notice the difference in electron charge ... Calcium is a divalent anion (++) but sodium is monovalent (+). this means that when the calcium is exchanged, TWO sodiums must replace ONE calcium or magnesium. the net effect is that while you are softening your water with respect to your laundry and those white spots on your faucet, you are actually INCREASING the total dissolved solids of your water.
the term 'hardness' arose historically because water with higher levels of Ca and Mg are more difficult to use for washing clothes, i.e, - it's harder to produce a lather from soap (or from detergent), hence “hard” water.
General Hardness (GH) is a measure of the concentration of calcium and magnesium ions (Ca++, Mg++) in your water. common ion exchange resins, both those used as pillows in tank filters and in bulk in household water “softeners” most typically exchange (Na+) for (Ca++) and (Mg++). the nature of resin chemistry dictates that the charges must balance, thus two (Na+) ions must be added to the water for each (Ca++) or (Mg++) removed. the resulting water will be 'softer' by the laundry definition but not by fish standards since the water now contains more total ions than it did before the softener.
however, the total dissolved solids (TDS) in the water are higher than that with which you started despite the fact that the GH test shows lower readings. the importance of this for fish-keepers is that the fish which we call "soft water fish" do not want "soft water" as we define it since fish don’t do laundry. they DO want water with low TDS which would include low GH as we measure it and also low Na+ and Cl-.
Amazonian fish and some of the SE Asians and riverine African cichlids come from waters with low TDS. the bottom line is that If you want lower TDS water, you must use peat extraction, de-ionization (DI), or reverse osmosis (RO). Otherwise you do not have appropriate water for "soft water fish".