I've only just started doing this and most of the time they are reasonable, for example I've had 6.9, 7.9mmol/L which is really fustrating at the time the alarm goes off below 5.0 (it may read something like 4.3 or 4.5 when it actually recognises the level). Let me know if you have any advice, I would set my libre alarm lower but if I set it to 4.5 I find I wake up with a crashing headache and genuine hypoglycaemic symptoms which ruins the whole next day for me headache wise - because by the time I'll have registered maybe I am bordering on ~4.0 (ish) territory.
Unfortunately the only solution here is to use something other than the libre app, which can apply fingerprick calibrations to correct the values that the libre is producing. Apparently some people get decent readings and can use the non-corrected libre app, I don't, I almost always have an offset, I therefore use the combination of Juggluco + XDrip+. There are other options, especially if you use an iPhone rather than an Android phone like me. Lots of threads about how to set things up if you decide to look at this.
However, stopping the overnight drop, even if it's not always actually to hypo level, would also remove the alarm issue. The question I'd try to answer is whether the early morning hypos are caused by your pancreas reacting to what you've eaten, or by the insulin you're injecting (i.e. too much).
How many carbs do you eat for supper and at what time? Can you post a graph showing your typical BG over this period through till you go low early morning?
I would certainly ask for a pen that can deliver 0.5U, I use one and it's very useful.
This won't last forever, eventually your pancreas will stop "helping" you, though from what I've read (as I don't remember it was so long ago), it can take quite a few months.