ClearAll["Global`*"];

ks = {{-2, -1}, {-2, 1}, {-1, -2}, {-1, 2}, {1, -2}, {1, 2}, {2, -1}, {2, 1}};

chCorners[kp_] :=
  Switch[kp,
    {2, 6}, {{2, 6}, {1, 8}, {3, 7}},
    {3, 7}, {{3, 7}, {1, 8}, {2, 6}},
    {6, 2}, {{6, 2}, {8, 1}, {7, 3}},
    {7, 3}, {{7, 3}, {8, 1}, {6, 2}},
    {6, 7}, {{6, 7}, {8, 8}, {7, 6}},
    {7, 6}, {{7, 6}, {8, 8}, {6, 7}},
    _, {kp}];

tryNext[p1_, p2_] :=
   Module[{path1 = p1, path2 = p2, lp, ns1, ns2, np1, np2},
     count = count + 1;
     ns1 = Table[Last[path1] + ks[[i]], {i, 1, 8}];
     ns2 = Table[Last[path2] + ks[[j]], {j, 1, 8}];
     If[count == 984, Print[Length[path2], " a ", path2]];
     Do[
       If[
         (! done) && (0 < ns1[[i, 1]] < 9) && (0 < ns1[[i, 2]] < 9) &&
          ! (MemberQ[path1, ns1[[i]]] || MemberQ[path2, ns1[[i]]]),
         np1 = Join[path1, chCorners[ns1[[i]]]];
         lp = Length[np1] + Length[path2] - 1;
         If[(count == 984), Print[Length[path2], " b ", path2]];
         done = (lp == 64) && MemberQ[ks, Last[np1] - Last[path2]];
         If[lp < 64,
           Do[
             If[
               (! done) && (0 < ns2[[j, 1]] < 9) && (0 < ns2[[j, 2]] < 9) &&
                ! (MemberQ[np1, ns2[[j]]] || MemberQ[path2, ns2[[j]]]),
               np2 = Join[path2, chCorners[ns2[[j]]]];
               tryNext[np1, np2]],
             {j, 1, 8}]]],
       {i, 1, 8}]];

done = False;
count = 0;
(* Starting two paths from lower left corned, {1,1} *)
tryNext[{{1, 1}, {2, 3}}, {{1, 1}, {3, 2}}];