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I'm not sure which points you really want, so this is a stab in the dark: You could "walk around" the inner resp. outer circle, and pick the closest point in pts to every point on each circle. Mathematica's Nearest function makes this relatively quick:

ADD: We can also connect each point on the inner/outer circles to it's closest point, to make the idea clearer:

Graphics[
 {
  Point[pts],
  Blue, {Thick, Line[pts[[ptIndices[rMin]]]]}, {Dashed, Opacity[0.6], 
   Circle[{0, 0}, rMin]},
  {Opacity[0.5], 
   Line /@ Transpose[{ptsOnCircle*rMin, 
      pts[[(nf /@ (ptsOnCircle*rMin))[[All, 1]]]]}]},
  Red, {Thick, Line[pts[[ptIndices[rMax]]]]}, {Dashed, Opacity[0.6], 
   Circle[{0, 0}, rMax]},
  {Opacity[0.5], 
   Line /@ Transpose[{ptsOnCircle*rMax, 
      pts[[(nf /@ (ptsOnCircle*rMax))[[All, 1]]]]}]},
  }, ImageSize -> 600]

To control the "jerkiness" of the lines, you can use a transform the "squashes" the points to a thinner ring:

I'm not sure which points you really want, so this is a stab in the dark: You could "walk around" the inner resp. outer circle, and pick the closest point in pts to every point on each circle.

(code for the animation at the bottom of the answer.)

Mathematica's Nearest function makes this relatively quick:

To control the "jerkiness" of the lines, you can use a transform the "squashes" the points to a thinner ring:

Here's the code for the animation at the beginning:

Monitor[frames = Table[Graphics[
     {
      {AbsolutePointSize[1/300], Gray, Point[pts]},
      MapThread[
       Function[{r, col},
        Module[{nearest, poly},
         nearest = nf[ptsOnCircle[[i]]*r][[1]];
         poly = 
          Append[TakeWhile[ptIndices[r], # != nearest &], nearest];
         {              
          col, {Line[pts[[poly]]]}, {Dashed, Opacity[0.6], 
           Circle[{0, 0}, r]},
          {Thick, Line[{ptsOnCircle[[i]]*r, pts[[nearest]]}]}
          }]], {{rMin, rMax}, {Red, Blue}}]
      }, ImageSize -> 300], {i, 1, Length[ptsOnCircle], 5}];, i]

ListAnimate[frames]

ADD: We can also connect of point on the inner/outer circles it's closest point, to make the idea clearer:

ADD: We can also connect each point on the inner/outer circles to it's closest point, to make the idea clearer:

To control the "jerkiness" of the lines, you can use a transform the "squashes" the points to a thinner ring:

ADD: We can also connect of point on the inner/outer circles it's closest point, to make the idea clearer:

Graphics[
 {
  Point[pts],
  Blue, {Thick, Line[pts[[ptIndices[rMin]]]]}, {Dashed, Opacity[0.6], 
   Circle[{0, 0}, rMin]},
  {Opacity[0.5], 
   Line /@ Transpose[{ptsOnCircle*rMin, 
      pts[[(nf /@ (ptsOnCircle*rMin))[[All, 1]]]]}]},
  Red, {Thick, Line[pts[[ptIndices[rMax]]]]}, {Dashed, Opacity[0.6], 
   Circle[{0, 0}, rMax]},
  {Opacity[0.5], 
   Line /@ Transpose[{ptsOnCircle*rMax, 
      pts[[(nf /@ (ptsOnCircle*rMax))[[All, 1]]]]}]},
  }, ImageSize -> 600]

To control the "jerkiness" of the lines, you can use a transform the "squashes" the points to a thinner ring: