Chance to find HMAC key/salt when having knowledge of the hashed data? I'm not sure how to word this... I'm working with a HMAC (I think of it being a "salted hash"). I know the entire string being hashed, I do NOT know the salt. I also know the first 8 characters of the resulting hash. I can also, repeatedly, generate new hashes by changing the input string, and retrieve new first-8 characters.

Does that in any way make it even remotely possible to find the salt? Even via brute forcing, would that decrease the entropy at all, if I were able to modify the hashing algo (much like how bitcoin is mined by pre-computing most of the hash and only updating the nonce at the end, removing many many steps from the hashing itself)?

From all I've learned, I'm leaning towards a "no", that its no help whatsoever. That being said, I've never worked with a HMAC/salted hash before, so I'm not sure if that works any differently. Is finding the salt any easier because of how a HMAC is computed?
Assuming you mean a normal HMAC with a strong unknown key, no amount of MAC values for known, or even chosen messages would help you find the key. That would mean a failure of HMAC's forgery resistance, which is expected of MAC algorithms.

(It would be a very bad case of forgery attack, since it would allow forging MACs for any messages. Even much lesser attacks, like being able to forge a single valid MAC for another message, would be considered significant breaks.)