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 and Sacks' Higher recursion theory are good resources on this topic, and are freely-and-legally available