It's similar like my idea, but I can't understand how you can calculate the multiplier.
You try to get the temp value from the temp value. I assume you use the old temperature value and the old known temperature to calculate the multiplier for the new one. Is it right ?
Pretty simple actually. How I am doing it is I go to a calibration screen on my LCD Keypad shield (DFRobot). I use the left and right arrow buttons to enter the known temperature (which I am getting from a manual thermometer). The LM35 is in measuring the same thing at the same time. After I enter the temperature, the multiplier is computed from the entered value and the actual value being read from the analog pin.