So, start at the highest point, follow each of edges of each of the triangles that contain that point, recursively ... to what end? Where am I going?...

I have no idea how efficent this will be; probably, the official literature will get you down a better path. However, my thought:

1. Make a subset of all the points that are connected to the top point (directly or indirectly) that are within deltaz from the top z_t. (So you might end up going down a few triangles, but stop if any of the z are less than z_t-deltaz)
2. sort the subset by each point's angle from z_t; example:

   $top = [$x_t,$y_t,$z_t]; # you got this from your maxima search
   @subset = ( [$x0,$y0,$z0], [$x1,$y1,$z1], ...); # the subset in step1
   @contour = sort { 
     atan2($a->[1] - $top->[1], $a->[0] - $top->[0]) 
     <=> 
     atan2($b->[1] - $top->[1], $b->[0] - $top->[0]) 
   } @subset;
   [download]

3. that sorted list is now the contour line -- every point in that list is within the same height band, and it is sorted such that you could draw a counterclockwise line through those points as your "contour", or calculate other angles as needed along the same contour.