Travel question when green card and passport have different names

My wife and I are greencard holders. My wife's passport expired recently. She also had her name legally changed after marriage. Her old passport had her maiden name and her new one has her current name. However, her green card has her maiden name. We have to travel to India urgently and I was wondering whether we will have any trouble re-entering into the United States. We have possession of the old passport (on which her name matches with her name on the green card) and marriage certificate .
You will need to deal with different entities: the CBP, TSA, and the airline.

Please see CBP Website:

Lawful Permanent Residents (LPR)
: LPR's who change their name due to marriage or because of any other circumstance may travel using your LPR Card (US Green Card; I551) in your prior name provided you bring proof of your name progression such as; a marriage certificate, a divorce decree or court documents showing a legal name change.

Please check with the country(s) you are traveling to for their entry requirements.

Please call Transportation Security Administration (TSA) about their requirement for name match on travel documents.
In the U.S. a middle initial on one document, when compared with other documents, does not carry any legal force. you may required a marriage certificate and permanent Address proof and then complete all legal formalities.