As SEJPM said, one of the common security assumption for a hash function is that there is no way to retrieve the hash but to calculate the whole function. In other word there should be no short-cutting.
Let us have a look of how we could accelerate calculation. This could be done either by trying to reduce the numbers of iterations over the message and by tweaking the main function.
Considering the 2 mains constructions schemes we have two cases:
- in a Merkle–Damgård construction (MD5, SH1, SH2) the compression function is composed of multiple iterations of a round function.
- in a sponge construction, the main permutation is also composed of multiple iterations of a round function.
In both case the message has to be completely absorbed in order to retrieve the hash. So no way to cheat on that part, we will need to process the whole message.
Why is this necessary ? Because each bit of the message has an influence on the hash. If it wasn't the case it would pose some serious security threat: the difficulty to forge a hash would be less complex.
However, we know that a hash function is composed of a round function applied multiple times. Can we simplify this process and have a tweaked round function that compute only the desired bits ?
It could be an idea, but you will notice that after few rounds, the bits that you are targeting depends on full states. E.g. for MD5, $19$ rounds (over the $64$) implies a full propagation of a modified bit (source code). Therefore the you need to compute the exact full state for the $64 - 19 = 45$ iterations left of the round function.
Therefore you during the main absorption phase you won't be able to cheat and only after fully absorbed the message, you will have to do $45$ normal rounds before using your $19$ tweaked ones. This surely is not a significant gain of speed.
Also notice that the number of operations (without considering the data provenances) will be relatively similar.