Some alternatives:

my $first = 1;
for my $n (sort { $hash{$a} <=> $hash{$b} } keys %hash){
   if ($first) {
      $first = 0;
      next;
   }

   ...
}
[download]

my @names_by_val = sort { $hash{$a} <=> $hash{$b} } keys %hash;
shift(@names_by_val);
for my $n (@names_by_val) {
   ...
}
[download]

my @names_by_val = sort { $hash{$a} <=> $hash{$b} } keys %hash;
for my $n (@names_by_val[1..$#names]) {
   ...
}
[download]

A hash isn't needed at all.

my %hash;
foreach my $n (@names){
    my $r = int(rand($range))+$min;
    $hash{$n}=$r;
}
my @names_by_val = sort { $hash{$a} <=> $hash{$b} } keys %hash;
[download]

can be replaced with

use List::Util qw( shuffle );
my @shuffled_names = shuffle @names;
[download]