Well, now I have a Visa card.
It's not, strictly speaking, a credit card -- it's a debit card. But it carries the Visa symbol, and the accompanying literature explains that "it's welcome wherever you see the Visa sign".
Can anyone who has attempted to use a Visa debit card in the USA confirm whether or not it's accepted by such organisations as car hire companies? I've already established that it can be used to purchase stuff on websites that don't accept my former ("Switch") card.