I don't think it will work for minus temps the way you got it.

Your output as I said earlier, would be directly from the LM35, meaning from the output pin AND the GND pin of the LM35. NOT the circuit Ground.

Now, that leads us to another problem. Your new output will be 0 volts at 0 C, positive at positive temps but negative below zero (LM35 output would seem more negative than the LM35 ground). So the opamp would not be able to handle this negative voltageas you designed it.

You are going to need a differential amplifier, such as the first opamp in my design.

Ioannis