Sounds like you are in the same situation I was in a year ago. My wife has also just come to America about 13 months ago. Once we got her SSN card, we applied and she was able to get a Capital One Card with a small CL $500 (Orchard turned her down as well). I believe Capital One has a program for people who just immigrated to the U.S.
I also added her as a AU on my credit card, and since then she gets CC offers left and right.
Her credit score is already around 720 and she now has a Gold Amex, Wamu card, Citi, and a Chase all in her name alone. She was able to get these cards about 6 months after her first card. We never had to send any information in. I do believe it takes about 3 months after she is added as a AU or is able to get her own TL before she can view a credit score.
Hope this helps