The big banks are able to resell banking services to the smaller ones and a lot of smaller ones sign up with those services. I work for a large bank and we process transactions for hundreds of smaller banks and are required to have redundant data centers which are seperated by something like 3000 miles.
A lot of companies were/are paying their employees their normal salary via direct deposit so even if they have evacuated they can still find a working atm somewhere and access their accounts. The people with problems are the ones that still get paper checks because its not as easy to reroute a check in the mail or send to a different location just by someone calling in and asking it to be rerouted, my wife is in payroll and they are having a hard time verifying peoples ID's who want a check mailed to somewhere other than their home address so make sure you have direct deposit as well as some cash on hand.