my $thispiece = {%$piece};  # create a deep copy

Be aware that {%$piece} creates a shallow copy. It may appear to be deep, but only because $piece is shallow to begin with, i.e., it has only one level. Try your code with a nested structure for $piece:

Win8 Strawberry 5.8.9.5 (32)  Fri 05/07/2021  7:11:33
C:\@Work\Perl\monks
>perl -Mstrict -Mwarnings

my $Action;

my $piece = { this => { that => { foo => { bar => 123 } } } };

foreach my $other (111 .. 113){
    my $thispiece = {%$piece};  # create a SHALLOW copy
    $thispiece->{theother} = $other;
    push(@{$Action}, $thispiece);
}

use Data::Dumper;
print Dumper($Action);

^Z
$VAR1 = [
          {
            'theother' => 111,
            'this' => {
                        'that' => {
                                    'foo' => {
                                               'bar' => 123
                                             }
                                  }
                      }
          },
          {
            'theother' => 112,
            'this' => $VAR1->[0]{'this'}
          },
          {
            'theother' => 113,
            'this' => $VAR1->[0]{'this'}
          }
        ];
[download]

Of course, if you're only dealing with a shallow structure in $piece to begin with, your code will be fine.