This process only needs to identify 1 word in the list that meets the criteria resulting in a unique list of the words that were contained within other words.

Then add a single line to my solution above and it should reduce the time taken by roughly a factor of 10. Ie. A projected 1 hour:

#! perl -slw
use strict;

my @words = do{ local @ARGV = 'words.txt'; <> };
chomp @words;

my $all = join ' ', @words;

my $start = time;
my $n = 0;
for my $i ( @words ) {

    for my $j ( $all =~ m[ ([^ ]*$i[^ ]*) ]g ) {
        next
            if $j eq $i
            or $j eq "${i}s"
            or $j eq "${i}'s";
        print "$j contains $i";
        last;                     ## Added
    }
}

printf STDERR "Took %d seconds for %d words\n",
    time() - $start, scalar @words;
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