I wouldn't look to a variable number of fields in the main table
. What you probably want here is a relational table, joined on some unique id. Using your example:
Companies and Staff, having a Company record with Company details, and with several Staff records attached to it.
Type can be 'Fax','Tel','Skype','Mobile','Email', etc
<select name="type" id="type">
Store the types in their own table, editable by admin, so you can use an integer value for the type - this becomes very important in terms of performance when you go to do joins and searches. Alternatively, this could be a simple associative array somewhere, but that's not very user-manageable.
So the staff member can have unlimited contact types, and are joined on user ID. This can make for more complex joins, but it's not too difficult.
Note that you'll never have to physically store the textual value of "type_id" anywhere but in the types table/array. It's just referenced by an ID. So any time that type changes, your database requires no updating, the displays change with it.