Consider what I like to call the CONSISTENT GUESSING problem.
Given as input a description of a Turing machine $M$:
If $M$ accepts on a blank tape, you have to accept.
If $M$ rejects on a blank tape, you have to reject.
If $M$ runs forever on a blank tape, then you can either accept or reject, but in either case you have to halt.
(Of course this isn't quite a language, but more like a computability analogue of a promise problem.)
Now, by a modification of Turing's original proof, it's quite easy to show that CONSISTENT GUESSING is undecidable (I'll leave that as an exercise for you).
On the other hand, it's also possible to show that there's no reduction from the halting problem to CONSISTENT GUESSING---i.e., that it's possible to construct an oracle $A$ that returns the correct accept/reject answer for every halting TM, but whose answers for the non-halting TMs kill off every possible reduction from the halting problem to $A$. Thus, CONSISTENT GUESSING should really be seen as intermediate in difficulty between computable and the halting problem.