That way you might just add a single 10ct component and get rid of the problem completely.
Just enough to raise the ground of the LM35 above the input sensitivity threshold.
One Si diode would probably work, two Ge in series would be slightly better (because you get more range).
So that would be:
| lm35 +---- out
The way it's drawn is not the way it is built, the diode would normally sit near the computer end of things, not right on top of the sensor.
The thermal drift of the other circuit is at least as large as that of a diode at room temperature, probably a whole lot larger.