(c) I bought the car last Tuesday and the day after it broke down.
This could also be seen as a sentence fragment - an incomplete sentence. Its as if the entire sentence should be:
"I bought the car last Tuesday and the day after it broke down some surprising thing happened".
Of course most people will infer from the context that you mean the day after you bought it, but it is an ambiguity that should be eliminated by constructing a better sentence.
I would prefer; I bought the car last Tuesday, and the next day it broke down.
But this doesn't help you in your quest for a sentence that uses "the day after", but I think that phrase will almost always be an incomplete fragment, and, as Mike says (and as every US high school English teacher will say) it makes me think... "the day after what?"