It looks like you want something like this:

my %hash;

while ( <INFILE> ) {
    ( my $line = $_ ) =~ s/\s+$//;
    my ( $Rank, $id, $C, $Position ) = split /\t/, $line;
    next unless $Rank =~ /\d/;
    $hash{ $Rank } = "$id\t$C\t$Position";
    }

while ( <INFILE2> ) {
    ( my $line = $_ ) =~ s/\s+$//;
    my @b = split /\t/, $line;
    next unless $b[ 0 ] =~ /\d/;
    my $Start_rank = $b[ 1 ];

    if ( exists $hash{ $Start_rank } ) {
        print join( "\t", $Start_rank, $hash{ $Start_rank }, @b[ 2 .. 
+9 ] ), "\n";
        }
    else {
        print "no match\n";
        }
    }
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