This is a great question. There are many different kinds of measuring body temperature, from oral/anal digitals, temporal/infrared digitals, glass, strips, non-contact infrared, and ear thermometers.
First off, any battery-operated thermometer is useless when the batteries run out. This can happen when the batteries wear down, but it can also happen when they are cold. On the other hand, alcohol and mercury based thermometers are delicate, and the mercury ones are very accurate (if hard to come by). Also, the digitals tend to be big and bulky, and when you're looking to lighten the load on an outdoor excursion, this isn't what you want. What's more, hypothermia is a condition of the core, not of the head. So an oral thermometer of any type will not be particularly helpful. Anal thermometers are, well, not very convenient to use, and would be difficult to keep sanitary, especially for communal use.
In our group, we have adults and children (we are boy scouts), and so anal readings are categorically ruled out, but, we do keep several thermometers that can be used orally or for the armpit. We also keep oral strips on hand. We rarely use them, though.
There are stickers or paper-based ones, but, my doctor tells me they're good for measuring high temperatures, not so good for measuring low temperatures. Keep and use them if you suspect fever, that reduces need to use the more delicate thermometers, which have sanitary, breakage, and loss considerations. Also, you can keep a ton of them and they compact well and weigh less to nothing.
Having said that, if you suspect hypothermia or hyperthermia, the last thing you want to do to someone (you certainly won't be able to administer to yourself, you won't have your wits about you) is to be poking and prodding with a thermometer. If you have the slightest thought about hypo/hyperthermia, assume it has set in and then administer first aid, regardless of what any thermometer might read. As mentioned in the previous answer, knowing the symptoms is much more accurate.
In my 4 decades of camping and in marching in parades, I've overseen over a hundred cases of hypothermia or hyperthermia, and in not a single case did we worry about temperature: we just treated for it and moved on. Even in a health lodge where there are quality thermometers around to take accurate readings, we simply treated regardless. There isn't more you can do if the thermometers confirm what you already suspect, whether it is hypothermia or hyperthermia.
If you really wanted to experiment, you'll need to spend big bucks - around $1000 - and buy or borrow a Heitronics infrared thermometer, which they make scientifically accurate. This can be used to confirm what you already suspect, but the cost just isn't worth it, unless you are a doctor or a hospital where confirmation is necessary. We once treated a man for hypothermia when it turned out he was drunk. No harm in that case, but there could be other conditions - like hypo or hyperglycemia, or other conditions, which can be ruled out with expensive equipment.
But when you're in the outdoors, best to stick with knowledge and intuition, treat, and if possible, get to a doctor to rule anything else out if needed.