*To move this one off the unanswered queue:*

No, the halting problem is much too weak to do this. Determining whether a computable linear order is a well-order is $\Pi^1_1$-complete, and in particular strictly more difficult than any computable iterate of the Turing jump (see "hyperarithmetic hierarchy").

Both Hinman's [*Recursion-theoretic hierarchies*](https://projecteuclid.org/eBooks/perspectives-in-logic/Recursion-Theoretic-Hierarchies/toc/pl/1235415519) and Sacks' [*Higher recursion theory*](https://projecteuclid.org/ebooks/perspectives-in-logic/Higher-Recursion-Theory/toc/pl/1235422631) are good resources on this topic, and are freely-and-legally available