Ha ha, my neighbor is a scientist at NIST. His job, literally, is measuring stuff. They have a piece of gear that's good for measuring temperature, guaranteed to be within something like a few thousandths of a degree. They can't move it without voiding the calibration, though, so you can't use one of those.
What you're discovering, BTW, is a joke that he likes to make. "Measure once, and you feel like you know the truth. Measure twice, and you realize you have no clue."
I have a couple of calibrated thermometers that work pretty well, but the good ones aren't cheap. The one I really like is a thermocouple connected to a Fluke multimeter. The Fluke was like $75 on eBay, and the TC and attachment were another $20. For as much as I've used it, it's a great price.
You can verify that a thermometer is "close enough" in this range, BTW, by putting it in a bowl of ice water. By definition, that water is at 32F. If the thermometer tells you differently, it's lying. Unfortunately, the magnitude of the error doesn't tell you anything useful: it could be either bias or gain, either of which will be indistinguishable for a single data point.
So to actually calibrate the thing, you'd need a second data point: boiling water. Tragically, a cheap thermometer may not be linear across that whole scale, so you still don't know anything.
Oh! And I should note that it seems pretty normal for a fridge to be non-uniform in temperature. So putting the thermometer in different places should produce different results.
Sorry, no help here, outside of "Get a real thermometer, dude."
Good luck.
Dan