You can use Replace field contents. One of the options will place a serial number in the current field.
If you open Field Definitions and set that serial field to auto-enter the serial (and don't worry whether it is the correct 'next number' or not), then when you run Replace Field Contents with 'serial', be sure to show all records first. Then set the starting serial to 1 and check the checkbox to 'update auto-enter serial. FM will then run the new serials through your table plus set the auto-enter serial correctly to the last serial it used + 1. You'll then be primed for adding new records in the table.
If you don't set your serial in Definitions to Auto-Enter serial FIRST then 1) Your Replace Field Contents will not offer the update next serial option and 2) new records created will NOT increment for you.