I wouldn't want to have zero Visa cards in my wallet due to the potential of Amex not being accepted.
Does "Visa Preferred" mean the Chase Sapphire Preferred? Or a different card? If you're trying to minimize your annual fees (swap Chase for Amex) you could consider getting the Amex Everyday Preferred, so all your earning is on Membership Rewards, but get a $0 AF Visa card as well (i.e. the 1.5% CB Quicksilver, or my preferred Visa from Costco if you are member, both with no foreign transaction fees).