As a general rule if you have no credit history in the US and you apply for a random unsecured credit card you will be turned down. That you have had an application declined won't appear in your credit report, but the fact that you applied will and this might have a small, temporary negative effect.
Of course your situation is a bit different in that, thanks to American Express, you will have some US credit history. I have no personal experience so I don't know if this by itself is sufficient for an application for another card to be successful, but if no one who's been in your situation has a more authoritative answer it might not hurt much to apply for a card you want once you are settled to see how it goes (and report back on the result?).
If that doesn't go well then you'll want to wait a while for your credit history to lengthen before trying again. If you really want a second card in the mean time and aren't fussy about exactly what card it is, a US bank which is a sibling of a UK bank you have a history with might be willing to give you an unsecured card based on your UK history if you open accounts with them (I know HSBC and Citibank sometimes will do this; I have had experience with HSBC for this in other countries). Otherwise a secured credit card is generally the only option for people with little or no (or bad) US credit.