I'd certainly consider a low power rail-to-rail op-amp and dump the 9V batteries.
Leave the opamp with it's ground connected to the joint between the two resistors, choose the bias so it drives the output from the -2.5 V the opamp sees (which really is the 0 from the computer) to the +2.5V (which is the +5 of the computer).
The reason why I think it matters is that batteries are a nuisance, they run empty and will cause a problem with long running experiments, so it's worth the extra time to engineer them out of the circuit.
From what I can see they're only used to power the op-amp.
The 'floating ground' trick is s.o.p. when designing op-amp circuitry that needs to be fed from a single supply.
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.