What are best known attacks for finding partial target pre-image?

Just a follow up on this question since I don't have enough point to comment: Does a partial preimage attack imply a preimage attack?

Partial target preimage attack: Given $M$ and t-bit partial target of $d \leftarrow H(M)$, find $M^*$ such that t-bit of $d^* \leftarrow H(M^*)$ is the same as the t-bit of $d$ at the same position.

My question is what best known attacks are.

In the generic setting, a $$t$$-bit partial preimage attack on an $$n$$-bit oracle $$m \mapsto f(m)$$ is the same as a full preimage attack on a $$t$$-bit oracle $$m \mapsto \operatorname{trunc}_t(f(m))$$. If $$f$$ was uniformly distributed in $$n$$-bit functions, then $$\operatorname{trunc}_t \circ f$$ is uniformly distributed in $$t$$-bit functions, so there's no advantage to be had in knowing that it is a truncation of a larger oracle.
Of course, for particular functions $$f$$, there may be better partial preimage attacks, like $$f(0^t \mathbin\| m) = 0^t \mathbin\| \operatorname{trunc}_{n - t}(H(m))$$, $$f(b \mathbin\| m) = H(m)$$. But we would need more particular details about $$f$$ like this to say.