If you phrase the problem as "design a better hash table, given the hash function h(x) = 4", then the obvious first thing to do is discard the provided hash fun

If you phrase the problem as "design a better hash table, given the hash function h(x) = 4", then the obvious first thing to do is discard the provided hash function, and use any reasonable hash table implementation with a new hash function, say h'(x) = x. (If x "is an object" instead of a primitive type; then some underlying representation of x in a register is a pointer; that memory address can typically be converted to an integer that can be the hash value.) Thus, I would argue that the problem reduces to the case where you have a reasonable hash function.