A late meal and bg was still climbing, injection site problems, insufficient insulin dose?......why not carry out some night time testing for a couple of nights to see what your bg levels are like throughout the night.
I wouldn't take any fast-acting insulin and would treat the hypo first and wait until my bg had returned to normal levels before injecting and eating my meal, never inject QA insulin if your hypo as this can be dangerous.
Maybe you went hypo during your sleep and the high blood sugar is the result of a liver dump. This happened to me many times when I was on injections. Night time testing to see what's going on with sugars and whether your basal rate is correct is definitely a good strategy.
You might sleep through a hypo and once your sugar level drops below a certain point your liver releases its own glucose stores into your bloodstream. This could account for high sugar levels in the morning.