Hi Sören,
so I understand your problem is specifically with the "coverfinger" (the finger tips) because of the small base surface, and it not sticking to the print bed?
The solution depends on various factors: What material are you using (PLA? ABS? Other?) What kind of bed (material)? Heated or cold? Are you using blue painter's tape, Kapton, ore glue stick?
On my end, I use PolyMax PLA, on a cold glass bed, with a thin layer of UHU stick. No support or raft needed, the parts stick so well to the bed that they are even hard to remove sometimes. Even with regular PLA, the finger tips are never an issue. ABS might be a lot trickier though - again, please specify your filament material and print surface.