If you are going to return a hash reference from extract_episode_data() ...

sub extract_show_info {
    my $input_string = shift();
    my $result       = undef;

    if ( $result = extract_episode_data($input_string) ) {
        $result->{type} = 'se';
    } elsif ( my @date = $_ =~ /$RE{time}{ymd}{-keep}/ ) {
        $result = { ... };
    }
    return $result;

}

sub extract_episode_data {
    my $input_string = shift();
    if ( ... )
    {
        my $episode_data = { season => $1, episode => $2 };
        return $episode_data;
    } else {
        return;
    }

}
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... why not set the type in there too? That would lead to something like ...

sub extract_show_info {
    my $input_string = shift @_;

    my $result = extract_episode_data($input_string);
    $result and return $result;

    if ( my @date = $_ =~ /$RE{time}{ymd}{-keep}/ ) {
        return { ... };
    }

    return;
}

sub extract_episode_data {

    my $input_string = shift @_;
   
    if ( ... ) {
        return { type    => 'se',
                 season  => $1, 
                 episode => $2
               };
    }

    return;
}
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