It may surprise you to find out that in the Hobby Lobby ruling, the Supreme Court did not actually rule on whether the government had a compelling interest in providing access to birth control.
This is what they said:
We will assume that the interest in guaranteeing cost-free access to the four challenged contraceptive methods is compelling within the meaning of RFRA, and we will proceed to consider the final prong of the RFRA test, i.e., whether HHS has shown that the contraceptive mandate is “the least restrictive means of furthering that compelling govern-mental interest.”
...
The most straightforward way of doing this would be for the Government to assume the cost of providing the four contraceptives at issue to any women who are unable to obtain them under their health-insurance policies due to their employers’ religious objections.
In other words, the conservative wing of the court has already accepted government-provided birth control as constitutional. I think it's a pretty safe bet that the liberal wing wouldn't have a problem with that either. Birth control isn't about to be denied to people by this court.