Following up on my earlier suggestion, if you were to invoke the program as script.pl files.* instead of as cat files.* | script.pl, then you would still be able to use the convenient <> operator for reading the input. You might try something like this:

while (<>) {
  # examine the first line of the file
  # do the appropriate thing with it
  # (for example:)
  if (/^Expense Report/) { $type = 'Expense' }
  elsif (/^Invoice/) { $type = 'Invoice' }
  else { die ... }

  # Now seek the file
  seek(ARGV, 0, 0) or die "seek: $!";
  process_data($type);  # call the appropriate processor
  # the rest of the data in the file has been eaten
}

sub process_data {
  my ($type) = shift;
  # call process_invoice or process_expense_report
  # depending on $type 
}

sub process_invoice {
  while (<>) {
    # do something
    last if eof;
  }
}
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The key part is the last if eof line. That tells process_invoice to return back to the main loop at the end of the current file. Without it, process_invoice would continue on to the next file automatically, because it is using <>.

The seek back in the main routine will ensure that the processor functions such as process_invoice see the entire file, including the first line.