So I concocted a scenario where a third party candidate won a presidential election with only 1.53% of the popular vote. To do as such I made the following assumptions:

  1. Every congressional district in the country is equally populous.
  2. Every congressional district has equal voter turnout.
  3. Only three candidates appear on the ballot in every state.
  4. Every voter votes for the same party for U.S. house as they do for president.
  5. Should no candidate receive the necessary 270 electoral votes, and the election is thrown to the U.S. house as laid out in the 12th amendment, then the Republican and third-party congressmen will want to work together to prevent the Democratic candidate from winning.
  6. Should assumption 5 be put into effect, the presidential candidate this coalition will throw their weight behind shall be from the party that controls more U.S. house state delegations.

So the scenario I put forth is not meant to be realistic, it's supposed to represent a logical extreme within the system. I'm well aware that assumption 1 is not actually true, as Montana, which has one congressional district, has almost twice the population of Wyoming, which also has one congressional district. As far as Texas being blue, I was just choosing the largest states for the democratic candidate, not necessarily states they would realistically win. The assumptions I chose were because I needed some ground rules in calculating the smallest percentage a candidate in a three-way race could win with, otherwise I could say something completely ridiculous like only one person bothered to vote in California. The way I made my calculation was as follows. I assumed the third-party candidate won one congressional district by one vote in the twenty least populous states. As there are seven states that only have one congressional district, the third-party would control seven U.S. house state delegations. I also assumed the Democrats controlled the 24 most populous U.S. house state delegations (if you're wondering how they can control the 24 most populous state delegations, but not actually hold all of those states on my electoral college map, the answer is gerrymandering). The next six most populous state delegations would be controlled by the Republicans. The remaining 13 state delegations would have split control (no one party controlling a majority of its seats). Since the third-party would control seven delegations, and the Republicans would control six, they would throw their weight behind the third-party candidate. Including DC, there are 436 congressional districts in the country of which the third party would have won 20 of them by a single vote over the other two candidates. 100/436*20/3 = 1.53%

This is not a logical extreme. It CANNOT happen because in order for a candidate to win each state, they must receive more votes than the other candidates in that state. You would have to have FAR MORE candidates on the ballot than just three (about 10 or so), AND voting would need to be spread out almost evenly between them all.

Um, yes it can happen? In the scenario I put forth above the states and congressional districts won by the third party candidate were won by only one vote. That means each candidate got roughly one-third of the vote, with the third-party candidate with just slightly more. You don't need ten candidates on the ballot for that. In fact, if there were ten candidates on the ballot, the smallest popular vote percentage a candidate could win with would be 0.02%

Actually, it could even be far less than that. In a close race if there's a faithless elector or a third party wins the 2nd congressional district in say Nebraska for example AND neither main party gets 270, the House would pick the next President from the candidates with 1 or more electoral vote.

Right, I forgot to list in my assumptions that I was assuming no faithless electors (because then technically the smallest percentage would be 0%). If you look through the scenario I put forth above, you'll see I already have it going to the House. I made the assumption that every voter in the race voted straight ticket (same party for president and the house). I made the assumption that the third party in the race needed enough support in the house in order for the republicans to throw their weight behind them to get elected, which is why on my map the third party has won several states. As you said, technically, to get considered by the house they only need one electoral vote, but in that situation their party wouldn't have enough support in the house to actually be elected. Also, this point doesn't really apply because I limited the number of candidates in the race to three, but, technically the house doesn't choose from all the candidates that have received at least one electoral votes, they only choose from the top three electoral vote winners.