How to Determine Existence of Turing Reducible Languages?
To narrow down the search space, consider what you know about closure properties and HPL. 


The class of decidable languages $R$ is closed against complement.
HPL is not in $R$.


You can conclude that $L \not\in R$;¹ otherwise, $\overline{L} \in R$ and therefore $\mathrm{HPL} \not\leq_T \overline{L}$.

On the other end, you need that $L \in RE$ (the set of semi-decidable languages) -- otherwise $L \not\leq_T \mathrm{HPL}$.

In combination, you need $L \in RE \setminus R$ which answers your question².




In particular, $L$ can not be finite since all such languages are decidable. The decider just checks a hard-coded table of all words in the language. With the same table you can also decide the complement.
Note how this implies $\overline{L} \not\in R$ which is enough for $HPL \leq_T \overline{L}$.