Based on the suggestions offered here and some discussions with other programmers, this is the working code I came up with, for anyone who is interested. Thanks for all the great suggestions!

sub merge {
    my @allparts = @_;

    # make an array of just the end points
    my @ends = map{ [$_->[0], $_->[-1]] } @allparts;

    # give us a starting point
    my @sorted = (shift(@ends));

    # while we still have unsorted ends
    OUTER: while(@ends) {
        # check each of the unsorted ends to see if they can
        # attach, we only compare them to the last element of
        # @sorted because if we have more than one element in
        # @sorted, then we already checked all of @ends against
        # the elements at the beginning
        for my $x (0 .. $#ends) {
            # see if we can attach it at the beginning
            if($sorted[-1]->[0] == $ends[$x]->[1]) {
                unshift(@{$sorted[-1]},@{splice(@ends,$x,1)});
                next OUTER;
            }   
            # see if we can attach it at the end
            if($ends[$x]->[0] == $sorted[-1]->[-1]) {
                push(@{$sorted[-1]},@{splice(@ends,$x,1)});
                next OUTER;
            }   
        }
        # if we get here, then we didn't find
        # any @ends that attach to anything
        # in @sorted, so we grab one of the
        # @ends to make a new section, and
        # continue
        push(@sorted,shift(@ends));
    }

    # now go through the sections in @sorted, and replace
    # them with the appropriate complete data from @allparts.
    for my $x (0 .. $#sorted) {
        my @ends = @{$sorted[$x]};
        my @full = ();
        # for each start/end combination
        while(my($start,$end) = splice(@ends,0,2)) {
            # find the chunk it came from
            #print "start=$start end=$end\n";
            foreach(@allparts) {
                #print "0=$_->[0]\n";
                #print Dumper($_);
                if(($_->[0] == $start) && ($_->[-1] == $end)) {
                    pop(@full); # get rid of the duplicate piece
                    push(@full,@{$_});
                    last;
                }
            }
        }
        $sorted[$x] = [@full];
    }

    return @sorted;
}
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