HMAC SHA256 - fixed vs dynamic key - security and collision impacts
No, it is not useful to change the key between invocations of HMAC.
That would not change the expected probability of collision, which remains about $n(n-1)/2^{257}$ for $n$ values under a model of HMAC-SHA-256 as a PRF. For $n=2^{40}$ (over a million million, the UK bilion), that's less than $2^{-177}$, the probability of 177 consecutive tails with a fair coin, that is negligible in practice.
That probability can be brought to zero using encryption of a counter with a block cipher.