Here is a recent article describing a simpler algorithm for sampling derangements. I also found slides for the presentation. Since the paper is so recent, I guess this means that a small modification of Fisher-Yates is unlikely to generate derangements, since someone would have already come up with it by now. Still, their algorithm is in-place and has better expected running time than retrying Fisher-Yates until you get a derangement.

Here is a Perl implementation I whipped up. It is slightly odd because I followed their lead and used array indexing from 1.

sub rand_derangement {
    my $n = shift;
    return if $n == 1; ## no derangements of size 1

    ## precompute $D[n] == number of derangements of size n
    my @D = (1,0);
    push @D, $#D * ($D[-1] + $D[-2]) while $#D < $n;

    my @A = (undef, 1 .. $n);
    my @mark = (1, (0) x $n);
    
    my ($i, $u) = ($n, $n);

    while ($u > 1) {
        if (! $mark[$i]) { 
            my $j = 0;
            $j = 1 + int rand($i-2) while $mark[$j];
           
            @A[$i,$j] = @A[$j,$i];
            
            if ( rand(1) < ($u-1) * $D[$u-2] / $D[$u] ) {
                $mark[$j] = 1;
                $u--;
            }
            $u--;
        }
        $i--;  
    }

    return @A[1..$n];
}
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