Instead of 6 different contact fields in your customer table, define a related table for this purpose where you have one record for each such "potential contact". Not only does this allow you to record from 0 to any number of such items, you can display this data in a portal on your layout--which can be inside a tab control if you want as well as still being able to open a new window to "pop up" a list of those records. And in FileMaker 13,you can put such a portal inside a pop-over or a slider as well.
Count ( RelatedTable::NeverEmptyField ) will return the number of such related records.
And if you decide to stay with individual fields, Count ( field1 ; field2 ; field3 ...) will count the number of those fields that are not empty.