I'm running into the same issue on the back of my home. It's very annoying. Once the ice dam is formed, icicles form, melt and drip water straight onto my steps. My solution will be a diverter. Something simple like a strip of ply wood, or another piece of gutter to simply "splash" the dripping water away from the steps. My entire second floor is brand new so I know it's not an insulation/heating/venting issue (normal cause for traditional ice dam's). My problem is that my white aluminum gutter stays a lot colder then the roof itself (Roof see's sun about 3-4 hours daily). Water runs into the gutter, hits the snow and gets stuck, Freezes then overflows and causes the mess.
Granite State Dyno and Tune
94 6speed Supra -- CT76TBB, CCW, Greddy, HKS, etc
11 Lexus CT200h -- Starfire/Caramel/Nav/Cam/Link/Audio