Job offer was rescinded due to incorrect info on background. Info was corrected. What now?

On December 1, 2016 my wife was offered a job in writing from hospital A. The offer letter states that the start date was to be January 9, 2017. It also listed the salary, benefits and that the offer was contingent upon a successful background and drug screen.

On the morning of January 6, 2017, she was contacted over the phone by the HR rep asking her if she had ever been employed by hospital B before. She confirmed for HR that she had been employed there 10 years ago when she was near the end of nursing school. Later that same afternoon, she got a call back from the HR rep saying that they were reviewing her background check and it was noted that she had been employed by hospital B and after contacting hospital B, it was confirmed that they had her listed as "not rehirable". Hospital B is a part of the same network as hospital A so they were not sure how or if they were going to move forward with the job offer.

My wife was not aware that hospital B and hospital A were part of the same group so on her application she marked that she had never been employed by the hospital group before. This hospital group still uses the name of a group they merged with several years ago which is not the same name that was asked about on the application.

On January 10 the HR rep and the VP of HR contacted her over the phone and told her that due to a combination of her withholding info on her application and being listed as not rehirable by hospital B, they were rescinding the job offer.
She immediately reached out to HR at hospital B and inquired about her not rehirable status. After much discussion, the HR rep sent it to their legal team for review. They determined that there was insufficient documentation and no approval for the not rehirable code to be on her personnel file so it was being removed.

HR at hospital A is being very tight lipped to this day and they have not mentioned giving her the job back that she was originally hired for. We are not seeking a lawsuit, my wife just really wants the job. Even after all of the run around, she is still in love with this job.

Do we have any recourse or should she walk away from the job and move on? They have told her that she can reapply for the job if they repost it, but there is no guarantee they would even give her a second look and she also feels like she would be reapplying for a job that is rightfully hers. Any advice?