I don't have exactly your situation, as I do have a partner who can stay home to supervise the children, but I do work full time, and do the majority of the homeschooling direct instruction. Of course, how much of what we do is applicable to you will be highly dependent on your circumstances, and the age/maturity of your children, but perhaps this can, at least, be a starting point for ideas:
I have a small amount of flexibility in my hours, which I've shifted to give us two afternoons a week at home, during which I work with our children. They spend one afternoon at a homeschool coop. The rest of the weekdays, they have task lists, which they work on independently (our oldest is an early adolescent, which makes this more realistic). I also check in with them in the evenings and on weekends.
If you can arrange for some kind of childcare during the day, maybe shift some of the extracurriculars to during your work hours, it might work to move instructional time to nights and weekends. There's no rule that says school has to be between 8 and 2 on Monday through Friday.
