Does there exist a undecidable infinite language with only a finite undecidable subset?
Let $S$ be the set of elements on which the decider $T$ answers MAYBE. We can construct a new decider $T'$ which correctly decides $L$ as follows:

If the input is in $S$, answer the hardwired correct answer.
Otherwise, run $T'$.

Hence $L$ is decidable.
This actually shows that if $L$ is undecidable, then every decider for $L$ must make infinitely many mistakes.