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I think Ulysses will have died from a coral snake bite, which was in his backpack/saddlebag and escapes the backpack after biting Ulysses. This snake's appearance leads to the horse's skedaddling and Ulysses being thrown off the horse, then the horse "rolls on top of him."

I say this based on a. the scarlet snake they showed (which is similar to a coral snake except it has noticeable white stripes versus the narrow ones on a coral - which has wide red/black stripes, not wide white ones), and b. Ulysses was seen as supposedly losing consciousness as he was riding the horse before he falls off and said horse rolls on top of him.