Is there an "additive" proof-of-work?
Your property of what I would call "improvable" is actually considered undesirable for PoWs. It would allow a prover to make continuous progress
toward a solution that satisfies the difficulty threshold. Which would
mean that a prover that's twice as fast as another, has more than double
the chance to win the race, which is considered unfair. Being twice as fast should just double your expected wins.

So in fact a PoW should be what is known as "progress free",
which would seem to require not having your property.