You may well be getting one, or the other, or a combination of both. The testing should help reveal.
The rebound effect should be the most obvious to discover and the most "fixable", because either lowering basal, or perhaps a snack like genie suggested, or a similar tactic or a combination, should allow you to completely avoid it.
The DP however is probably here to stay, a natural cycle of your body. There are of course things to try to overcome the high effect, but it certainly can be tricky to deal with on MDI.
So the first step is probably to find out/rule out/fix up the rebound
Have fun testing.
