Hundreds of financial institutions are jumping onto Apple Inc.’s new mobile-payment system, hoping to keep a tight grip on customers who might otherwise be tempted to switch banks if their current debit and credit cards can’t be used with the newest iPhone.

The scramble began as soon as Apple announced the new Apple Pay service last month, say industry executives. In its initial announcement, Apple included the nation’s largest banks, as well as credit-card networks Visa Inc., MasterCard Inc. and American Express Co. But the...