In that case this double data structure to grant you fast indexed access to sequences is overkill, and you should rather go with grandfathers design.

use strict;
use warnings;
use Data::Dump qw/pp dd/;
use 5.10.0;

#  ========= Tests

my $set = Some::Module->new(qw(foo bar bat));


say $set->contains('fox');   # false
say $set->contains('foo');   # true
say $set->contains('FOO');   # true

say $set->after('foo');      # 'bar'
say $set->is_last('bar');    # false
say $set->is_last('bat');    # true


package Some::Module;
use Data::Dump qw/pp dd/;
sub new {
    my ( $class, @sequence ) = @_;

    my $pos = 0;
    my %positions = map { lc($_) => $pos++ } @sequence;
    my $count = @sequence;

    my $obj =
      {
       sequence => \@sequence,
       positions => \%positions,
       count     => $count,
       last_pos  => undef,
      };

    bless $obj, $class;
}

sub contains {
    my ( $self, $key ) = @_;

    my $pos = $self->{positions}{lc($key)};

    $self->{last_pos} = $pos;

    return defined $pos;
}

sub after {
    my ($self, $key) = @_;

    my $pos = $self->{positions}{lc($key)};

    return $self->{sequence}[$pos+1];
}

sub is_last {
    my ($self, $key) = @_;

    my $pos = $self->{positions}{lc($key)};

    return $pos == $self->{count} -1;
}
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