Here's a way of thinking about the undecidability of the halting problem. Let's say you've got a person who's amazing at reading minds, and you bring someone off the street and tell them they can either have steak or a cupcake (but not both). You then ask the mindreader to decide if the person will have the cupcake or the steak. Conceivably, they might be able to figure out which one the person will have. Now let's say that you add a twist: you walk up to the person from the street and tell them what the mindreader predicted; in that case, the mindreader can't succeed because the subject can choose to do the opposite. That's similar to how the undecidability proof of the halting problem works.

Now instead of a mindreader, we have a halting oracle, and to make its job impossible we have a test program that is "made aware" of the halting oracle, and does the opposite of what the oracle says. Impossible problem for the oracle. But that then begs the question, how many potential applications of the halting problem will involve test subjects that actually know what the halting oracle thinks? How many test subjects even know about the halting oracle? For instance, how can a program that looks for counterexamples to the Goldbach conjecture know anything about your halting oracle? In these cases, the undecidability proof doesn't apply.

So the answer to your question is conceivably yes.