Once in the redemption period there is no other way for a homeowner to stay in their home unless they "redeem" on the property and pay the full balance of what was bid at the sale. This is not always what a person's mortgage balance was, so make sure to check the disclosure mailed to your home that states the balance bid.
So whether you can do a reverse mortgage, pay cash, win the lottery, or however else you want to pay the balance off that is it.
The only exception is the government like Fannie or Freddie that I have come across. The rules are changing constantly, and I have had clients who went back to a attorney and found out their file was mishandled, and indeed they did qualify for a loan mod. It never hurts to consult an attorney and in fact I always advise it. This is extremely rare and a lengthy process but I have seen it happen.
I noticed BoA is starting to publicly announce principal reductions for select homeowners. I hope anyone who reads this will understand that if you want to stay in your home pre-sheriff sale is the greatest opportunity. If you wait until the sale happens to contact a Realtor, counselor, attorney, etc... your options are greatly limited.