You seem to be onto something here. Using my latest results, try the following:
Table[{{n, n + 1}, Intersection[Round[100000 Sort[MyEuclidean[#[[1]], #[[2]]] & /@ Subsets[biggestlittlepolyhedra[[n]], {2}]]],
Round[100000 Sort[MyEuclidean[#[[1]], #[[2]]] & /@ Subsets[biggestlittlepolyhedra[[n + 1]], {2}]]]]}, {n, 10, 34}]
{{{10,11},{100000}},{{11,12},{100000}},{{12,13},{100000}},{{13,14},{100000}},{{14,15},{100000}},{{15,16},{100000}},
{{16,17},{100000}},{{17,18},{100000}},{{18,19},{100000}},{{19,20},{80881,100000}},{{20,21},{84172,100000}},
{{21,22},{88789,100000}},{{22,23},{37206,100000}},{{23,24},{66878,100000}},{{24,25},{100000}},
{{25,26},{64029,100000}},{{26,27},{69192,100000}},{{27,28},{35135,39229,59992,68950,69689,87104,88930,100000}},
{{28,29},{29935,86265,88761,100000}},{{29,30},{88650,89151,100000}},{{30,31},{89151,100000}},
{{31,32},{73929,91003,100000}},{{32,33},{87444,100000}},{{33,34},{51155,56063,90062,90206,100000}},
{{34,35},{38743,100000}}}
In these best known results, found by a random process, there is an improbable amount of overlap in the point distances between solutions for 27 and 28 points. All of these will have overlaps on the longest distance. But there seems to be some repeated structures within this data, something I hadn't noticed. See if you can map together the 27 and 28 point solutions.
Another exercise would be to take a solution for
$n$, remove two points, and try to get a solution for
$n-1$ by combining those points.