If you don't run on a *NIX platform or you don't have cat available for some other reason then something like this would do the job as well:

use IO::File;

my $output_fh = IO::File->new( 'fullstaffinfo', 'w' ) or die $!;

for my $input (<staffinfo*>) {
    my $input_fh = IO::File->new( $input, 'r' ) or die $!;

    $output_fh->print(do { local $/ = undef; <$input_fh> });

    $input_fh->close;
}

$output_fh->close;
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Cheers, Flo