Thanks to hdb, RichardK & pme.

Based on ideas from your code I've derived two routines:

1. The first follows the basic schema of picking $count evenly distributed points offset by $step/2 and adding rand( $step ) - $step/2;

   but except for the first and last, it extends the random adjustment to rand( $step * 2 ) - $step, which allows for occasional overlaps (out of order points):

   sub rndPart1{ ## can overlap
       my( $start, $end, $count ) = @_;
       my $step = ( $end - $start ) / $count;
       $start += $step / 2;
       map{
           my $mid = $start + $step*$_;
           my $v = $mid + (
               ( $_ == 0 || $_ == ( $count - 1 ) )
               ?  ( rand( $step ) - $step/2 )
               :  ( rand( $step * 2 ) - $step )
           );
           sprintf "%5.2f", $v;
       } 0 .. $count-1;
   }
   [download]

   Some samples:

   0 .. 59 / 10
    3.90 14.52 13.91 22.28 26.68 31.21 40.83 49.55 55.17 57.06
    3.87 12.36 12.22 26.06 25.67 31.14 37.76 47.42 52.80 55.30
    3.74  8.34 20.48 20.20 22.88 26.66 39.94 47.00 47.97 53.26
    4.44 12.14 11.63 18.69 31.22 30.80 42.69 39.36 48.64 57.32
    3.04 12.60 17.57 16.92 27.94 30.26 42.05 41.96 45.39 57.34
    5.74  3.39 12.46 23.94 20.98 29.66 34.13 46.07 49.56 54.50
    2.00  4.88 11.60 16.24 26.15 37.21 34.44 41.85 52.23 53.88
    4.87  9.13 17.46 24.54 32.38 35.39 36.27 48.09 55.00 56.48
    1.52  8.07  8.90 16.06 26.18 32.01 38.03 43.04 49.86 53.28
    2.75 14.32 11.62 15.78 22.40 35.69 44.05 38.89 49.32 54.03
    0.33  8.01 16.44 25.37 23.51 37.71 43.70 40.09 52.35 58.36
    4.54  6.33 19.35 18.14 31.89 31.81 42.15 46.19 45.40 57.71
    4.04 13.48 16.71 17.19 24.93 37.93 35.41 46.74 45.12 53.98
    4.50 13.99 12.20 19.43 30.19 28.36 36.90 43.51 44.91 54.97
    0.86  8.17 18.82 18.60 27.86 34.60 42.55 40.38 46.83 53.54
    0.52  4.29 19.80 15.06 31.78 32.97 35.28 44.16 52.55 57.26
    4.02 10.40 10.75 25.39 29.97 38.23 40.80 42.73 45.64 56.67
    3.80  8.54 12.36 25.85 20.68 30.68 40.63 38.98 52.05 57.25
    3.95 13.69  9.22 25.05 20.82 29.50 33.54 48.19 50.64 58.39
    0.41  6.58 13.65 25.90 26.83 33.76 34.38 46.55 45.82 55.61
   [download]

2. The second produces strictly ordered -- I found I needed that sometimes -- but it extends the range of the adjustment down to the last picked value, thus allowing for skewed distributions whilst favouring more even ones.

   That is, for args ( 1, 4, 3 ), the first pick is 1.5 ± 0.5 as normal, but if that pick is (say) 1.1, then instead of the second pick being 2.5 ± 0.5 it becomes 2.5 + (-1.4 .. +0.5); and if that picked (say) 1.2, then the third pick will be 3.5 + (-2.3 .. +0.5 ).

   The probability of extreme skewness decreases exponentially. The input parameters here are (1,100,10), and the count is the number of tries before the highest value falls below 90% (80%,70%...):

   E:\Chart>rndChart -L=90
    3.19  7.43 15.68 19.31 47.80 48.67 50.26 68.37 68.46 82.69 after 2 tr
   +ies
   
   E:\Chart>rndChart -L=80
    7.50 16.23 18.26 31.39 50.49 57.02 57.21 66.03 69.93 72.89 after 3 tr
   +ies
   
   E:\Chart>rndChart -L=70
   10.10 17.22 17.46 37.29 39.13 58.73 60.07 62.71 65.91 68.60 after 43 t
   +ries
   
   E:\Chart>rndChart -L=60
    9.50 18.43 19.91 24.09 47.55 48.76 49.03 53.83 54.86 57.29 after 67 t
   +ries
   
   E:\Chart>rndChart -L=50
    4.88 19.23 29.45 39.03 40.90 42.33 43.60 43.81 45.65 49.11 after 3253
   + tries
   
   E:\Chart>rndChart -L=40
    5.10 12.01 14.57 14.75 20.23 21.83 27.22 31.43 31.72 39.10 after 1000
   +17 tries
   
   E:\Chart>rndChart -L=30
   10.16 12.98 15.25 15.67 17.10 17.89 21.85 23.07 28.24 28.28 after 2980
   +335 tries
   
   E:\Chart>rndChart -L=20
   Terminating on signal SIGINT(2)
   [download]

   It currently only does negative skewness. (If anyone can see a way to also produce positive skews with a similarly exponential rareness, I'd love to hear of it?)

   The code>:

   sub rndPart2{ ## strictly ascending, skew possible
       my( $start, $end, $count ) = @_;
       my $step = ( $end - $start ) / $count;
       my $last = $start;
       $start += $step / 2;
       map{
           sprintf "%5.2f", $last = $last + ( rand( $_ + $step/2 - $last 
   +) );
       } map{ $start + $step*$_ } 0 .. $count-1;
   }
   [download]

   Some samples:

   <Reveal this spoiler or all spoilers in this node or all in this thread>