This is actually creating an array with one element, which is an array reference:

my @arr=['rose','orange','green'];
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What you actually want is to use parentheses, as I do below.

use Data::Dumper;

my @arr = ( 'rose', 'orange', 'green' );
my %hash;
$hash{'first'} = \@arr;
print Dumper \%hash;

my @newarr = ( 'red', 'blue' );
push @{ $hash{'first' } }, @newarr;
print Dumper \%hash;
__END__
$VAR1 = {
          'first' => [
                       'rose',
                       'orange',
                       'green'
                     ]
        };
$VAR1 = {
          'first' => [
                       'rose',
                       'orange',
                       'green',
                       'red',
                       'blue'
                     ]
        };
[download]

You might want to look at perldata, perlreftut, perlref, and References quick reference.