Not how it works due to FPTP in the states. Even if we look at the electoral map as of 2004, when it was easier for Republicans to win, if W. Bush had faced a popular, charismatic christian dominionist or something, even if that individual had been competitive in some deep-red states, in most of the states it would have led to Kerry winning as votes were sapped from Bush, Kerry would have taken the states he got, plus all of the swing states and won it in a walk, with precisely the same number of votes.
With a strong conservative-leaning third party run, all of the swing states become a lock for the Dems and Clinton cruises to the white house.
The election of 1824 was an aberration in this regard because it was a very different time, per how electors behaved and how the voting process worked, even though it technically operated under the same rules as today. But even if we were to entertain a scenario where the House could pick the next president, it would have to come from a third party candidate who drew evenly enough from Rs and Ds to be competitive across all states and not just play spoiler by sapping one side or the other.