Thanks guys! I didn't find "rail-to-railness" as a search parameter at Mauser or Farnell, but after some manual sifting through I decided to go with MAX985 family, that seems to be a good pick for a quite universal comparator. There was also MCP6541/MPC6546 family, but they are slower.
TLV3701 family is nice, thanks for the idea. Now I'll go with something faster, but I'll remember them when I'm focused on power consumption.
@Alti, thanks for the out-of-the-box answer. It'll help me use up the LM393's I already have.