I moved to United States last year in April 2012. I got my first Amex Gold charge card using Global Transfer, was approved. Currently I have 5 Personal credit cards and 4 Business cards from prime lender.
You will have better luck to get your next card from Amex in short timeframe, as they will give consideration based on your existing relationship with them.
Build your 6 months to 1 year credit history, then apply for more cards from Chase and other prime lender. If you consider to apply under 1 year timeframe, then consider to apply for cards that may not have minimum CL requirement of $5000.
But be very careful, do not apply too many cards at once. It may take 2-3 years to build stronger credit history.