No, you shouldn't use your hot water.
You didn't mention your tank size, but did mention 4, 1 gallon jugs, suggesting that would be enough for your water change. If I was you, I'd get a 5 gallon pail and an inexpensive submersible heater ($15 Tetra - Wallymart). Fill the pail with your cold water and set aside to come up to room temp. Then drop the heater in the day before your water change to get up to tank temp.
This solution is inexpensive and also gives you a temporary heater in the event your tank heater fails someday.
Edit: softeners use salt to strip/replace calcium, magnesium, potash... in favor of the sodium - not good for fish tanks! In this case only a portion would be used mixed with cold for warmth, but not worth the risk.
Last edited by AbbeysDad; 05-20-2011 at 08:19 PM..