Credit card recommendations?

Can anyone recommend a credit card company that will let you sign up with your legal (male) identity and billing address but issue that same person a (second) credit card with their female name? I'd like to be able to buy stuff while I'm out en femme without dealing solely in cash and without having awkward questions about the obviously male name on my current card. My current credit card's web site will let me request additional cards, but they want a social security number for each one, so I thought I'd ask here first in case anyone has advice.

Many thanks!

-- Michelle