I don't use humalog, so I'm not 100% sure but if you're having a hypo at 3-4 in the morning, I would have thought that's very likely to be the lantus, not the humalog. I have had similar issues where I was taking a plunge about five hours after taking my apidra. I stopped my morning basal and fixed the problem.
I would talk to your DSN but this sounds quite like my situation. I was getting quite a few hypos before meals as I'd upped my basal to take account of rising morning levels. Basically I get mild dawn phenomenon and my am levels are 6-8 most mornings, but if I test early morning, say 5am, they are good. In trying to get the am levels down I was taking too much basal, thus causing lots of hypos.
I was on lantus but switched to levemir and started 50:50 splitting but ended up on about 25%am 75%pm. I eventually dropped my am levemir as I mentioned earlier. I was doubtful when people said it here but hypos before meals are very likely to be caused by too much basal. The fast acting should be going out of your system by 5hrs plus, so shouldn't be causing a problem. I found a phase of more frequent testing (e.g. every hour between breakfast and lunch) helped me identify the pattern. If the rest of your levels are good, then getting 6.5 in the morning is not a problem, or certainly far less of a problem than hypos in the night.