Some suggestions:

• The beginning looks fine, no comments there.
• 'create hash of entire group' looks like it has a bit too complicated logic in the ifs in the innermost loop but the indentation is broken and I don't know the format your data is in, so it's hard to give advise there.
• Don't put the finished people into an array. Instead directly manipulate the hash. Just set the corresponding value to undef - instead of the push you then have $employee{$col} = undef;
• This obliviates the need for the delete loops.
• Instead you just use the following to print out the list:print LIST $_, "\n" for grep defined, values %employee;i.e. print to the filehandle LIST only the defined values of the hash %employee.

-- Hofmator