If you had JUST the sensor and you knew it's characteristics (for now just range of resistance), you can condition in a manner suitable for 0-3, by using something less than a 3V source.
Linearity, you will have to compensate for in your program R(x) = t(x).
Depending on the value of the sensor, a simple voltage divider can be used. You pay attention to the impedances in the circuit, so they don't introduce large errors.
The sensor is a variable impedance and so is your a/D converter. Thinks you can't do is measure a 1 Meg sensor with a meter with a 1 Meg input Z easily.
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Your system, as it stands:
1) probably has a resistor inside the temperature meter.
2) The sensor is a NTC thermistor.
Now, I haven't a clue what the voltage from the sensor to ground is, nor do i have a clue what the meter is doing. Nor do I know if your keeping the meter.
So, it requires some measurements: Suppose the measured voltage was 0-7 V across the sensor and the sensor was 100-500 ohms. We could divide the voltage by a little less than 2 and if we appropriately sized resistors, we don;t compromise accuracy.
Voltage sources we think of ideal. They have an output Z of zero. Meters, like your DVM probably has an input Z of 10 meg ohms.
10 megs in parallel with 0 is still 0, but 10 megs in parallel with 10 meg ohms is 5 meg ohms. The meter here affects our measurement.
If you plan to keep the original sender, then it may be possible to clamp the output, so when the sensor opens, the A/D doesn't see 12 V.
Hesitance vs temperature can be dome in a bucket with an ohmmeter and a thermometer.
We cannot do much of anything until:
1) Are you keeping the original gage?
2) Range of sensor? Area of interest? Accuracy required?
3) Input Z of A/D? Max voltage: (3V)? Ratiometric?
4) Voltage across sensor vs temperature especially if #1 is a yes.
5) Resistance vs temperature if #1 is a no.