Hippocrates' relevance to philosophy: medicine and philosophy closely related in Greek thought (as, in general, close relation between science and philosophy).
Like philosophy, (semi-)detached from religion--e.g. epilepsy, the "sacred disease", has natural causes, so no more or less sacred than any disease. No more, because epilepsy has natural causes. No less, because the natural causes are instances of divine principles. Note similarity to Xenophanes' denial of anthropomorphic gods.
Note we've had theories of nutrition (Anaxagoras) and drunkeness (Heraclitus) among the Presocratics, and Empedocles' idea that disease is an imbalance among the roots. Diogenes of Apollonia also has an Anaximenes-like theory of air as fundamental principle, explaining life by connection of air to breath (pneuma).
The Hippocratic corpus covers technical medical matters, philosophical foundations, medical ethics, etc. Question of whether foundation of medicine is experience or rational principles, i.e. derivations from the pre-Socratic schools.
But some general Hippocratic principles certainly seem philosophically derived--e.g. idea of balance and the four-humor (black bile, phlegm, blood, etc.) theory. Note that, eventually, these humors expand not just to explain disease but character (e.g. melancholy).
(Epistemic) modesty of Hippocratic corpus: reluctance to use invasive procedures; idea of medicine as art rather than application of rules.