In response to the Hobby Lobby decision, democrats in Congress are planning to prioritize a bill that overrides the Supreme Court's expansion of rights under the Religious Freedom Restoration Act (RFRA). According to Sen. Tom Harkin (D-IA), "The Protect Women’s Health from Corporate Interference Act reinstates the ACA’s contraceptive coverage and protects the right of all Americans, men and women alike, to make decisions about their medical care in consultation with their doctor, not their boss."
With no Republican supporters as of yet and vigorous opposition from major religious entities, the bill has very little chance of passing. It is likely intended rather to bolster voter turnout for Senate Democrats in tough races (e.g., Colorado and Alaska).
A different legislative response might have better chance of success. Instead of trying to overturn the specific decision, what if Congress worked together to make sure some of the "parade of horribles" potentially unleashed by Hobby Lobby can't happen?
For example, it is probable that in the near future a corporate RFRA claim will be brought challenging required vaccines and other immunizations. (At the state level, an individual case has already happened: Phillips v. New York.) Since the Court's majority assured us that these kinds of claims are not going to win because of Hobby Lobby, Congress could hold them to that. With the highest rate of measles outbreak in decades (presumably thanks to anti-vaxxers), now would be an opportune time to clarify that corporations cannot use RFRA to reject laws of general applicability executed by the Centers for Disease Prevention and Control (CDC).
One can also imagine a resource-extractive corporation bringing a RFRA claim challenging environmental regulations. Biblically-based principles of human dominion over the earth are certainly sincerely held by some companies that use mountaintop removal as a method. Now would be an opportune time to codify that corporations cannot use RFRA to reject laws of general applicability executed by the Environmental Protection Agency (EPA).
Not every federal agency would be as easy to get majority votes on. Labor, Education, and others would probably not be worth the effort. And maybe it's naive to think that Republicans would support even a bill that says corporations can't bring a RFRA claim against the CDC. But it would at least be an attempt with a legislative chance of success at marking up the slippery slope.