Correct! The cock was also a frequent sacrifice to Demeter's daughter Persephone. This famous clay plaque from the southern Italian colony of Locri depicts the underworld's First Couple at home. Notice the cock underneath Persephone's throne. Just why the bird was sacred to both the god of medicine and the queen of the dead is something perhaps best not thought about too much.
According to Plato, Socrates' last words were "Crito, we ought to offer a cock to Asclepius. See to it, and don’t forget." Scholars suggest that what Socrates meant by that was that he was just as happy to die as to continue living, or that this is evidence of his belief in the immortality of the soul (since Asclepius could be considered the healer of both body and soul). Others claim that Socrates was just praying to Asclepius to grant him a relatively easy and painless death, though promising a sacrifice after the fact might not sound like terribly good insurance.