I have spent considerable effort in finding a thermometer which would reliably and repeatably measure with 0.1°C accuracy.
Hi, I was involved in the same sort of thing early in my photo lab years, when I spent considerable time working in QC. We came to the conclusion that our best course of action was to rely on the Kodak Process Thermometer, the "best" they had to offer (at least to the general public). Kodak "Process Control Strips" would be the final arbiter of "correct" processing. (Since Kodak invented the C-41 process, they are the de facto authority on what "correct" processing is.) These strips are, in reality, the only way an outside processor can determine if they are within spec or not; the normal parameters - time, temperature, and agitation - are not enough for precise control. This is mainly due the difficulty of quantifying "agitation," but the temperature issues are another variable.
Anyway, when we needed to verify a processor temperature, someone would use a Process Thermometer held near that machine's temperature probe to confirm the digital readout, as best we could. After this, that machine's temperature controller would handle things. As a note, we kept three Process Thermometers in our chem lab (with restricted access) - if there was a disagreement, two out of three won the "argument."
I don't really want to advise others on how to proceed, but for a casual user, I'd say to buy the best thermometer that you are willing to pay for, then follow manufacturer instructions. If results are good enough for you, they're good enough. If you want to go a step farther, split a roll with a pro lab and compare their process results with yours (make sure you know how their control plots look, though). If you want to go farther yet, start working with control strips and a densitometer (a friendly lab might sell you some and read them for you).
I should say that I've never processed E-6, mainly color neg and printing. But if you had an "in control" (more properly said "in spec") C-41process as per control strips, then changed the developer temperature by 1/2 degree F, I'd bet a lot of money that your charts would be out of spec.
If we would have had a film processor that went an indicated 3/10 degree F off (on the digital readout), AND WE DIDN'T KNOW WHY (and didn't have assurances that it was coming right back), we would have put on a control strip and "leadered it down", meaning to shut down a production machine running 50 feet of film per minute. It was that serious to us. But we had specific needs with a finely tuned production system; most people would not be so finicky. Btw, it was a chain studio portrait lab, printing probably more paper volume than most people here could imagine.