@sleepwakecycle (Random internet person here who has used F.lux a lot)
Don't think their UI lets us do that. I have the same issue in my location. Using 8am allows an hour or two of less color shift before really mangling the colors. But 6am is a straight shot.
I agree you're not alone in wanting that much control. I'll bet night shift workers would benefit from being able to force changes outside of the normal day/night cycle (that they likely block outside light from windows anyway). Wish this was Open Source so users could add that.