So, I have been approved for a global transfer of my AMEX card from the UK to the US and I know it will take time to build credit rating/scoring. I am also wondering if it is as good idea to apply for another credit card? Will it harm or do any damage to my non credit rating at the moment or when credit does finally start to build?

The reason I would like another card is simply due to the benefits I can get (membership benefits) compared to the current AMEX card I have.