Have made a few modifications:

use strict;
use warnings;

my ( @header, %hash );
my @files = qw/File1.txt File2.txt/;
local $, = "\t";

for my $file (@files) {
    open my $fhIN, '<', $file or die $!;

    while ( my $line = <$fhIN> ) {
        my @columns = split ' ', $line;

        if ( $. == 1 ) {
            @header = @columns;
        }
        else {
            push @{ $hash{ $header[$_] } }, $columns[$_] for 0 .. $#co
+lumns;
        }
    }

    close $fhIN;

    for my $key ( keys %hash ) {
        if ( $key =~ /^dataS\d\dR\d$/ ) {
            print $key, @{ $hash{$key} }, "\n";
        }
    }

    undef %hash;
}
[download]

All columns are kept. After the script has processed a file's lines, it iterates through the hash keys. Note that a regex attempts to match the heading pattern for the columns you're interested in processing. Now, when there a match, it just prints the key and the associated list of values.