Fix the voting method first
There is no point improving the viability of third parties in U.S. politics before fixing the voting method.
This is because Duverger’s law states that there cannot be three viable political parties in a plurality voting system (which is currently the voting system for most elections in the United States):
Duverger presents the example of an election in which 100,000 moderate voters and 80,000 radical voters are to vote for candidates for a single seat or office. If two moderate parties ran candidates and one radical candidate ran (and every voter voted), the radical candidate would tend to win unless one of the moderate candidates gathered fewer than 20,000 votes. Appreciating this risk, moderate voters would be inclined to vote for the moderate candidate they deemed likely to gain more votes, with the goal of defeating the radical candidate. To win, then, either the two moderate parties must merge, or one moderate party must fail, as the voters gravitate to the two strongest parties.
In other words, a three-party system is inherently unstable, because moderate voters acting in their own enlightened self-interest will cannibalize one of the three parties to avoid suffering an electoral bloodbath at the hands of radical voters.
This implies that the two dominant political parties can remain fairly complacent. So long as each party is marginally better with respect to their agenda then they can trust that their supporters will suppress third parties in order to avoid undermining their agenda.
Now, suppose that a third party were to miraculously dethrone one of the two dominant parties; Duverger’s law would still apply in the wake of such an upset. The same forces would conspire to maintain the complacency of the two new dominant parties; the only difference would be the name of each respective party.
This is why we must reform our system in the right order: first fix the voting methods to mitigate the impact of vote splitting and only then can a third party meaningfully reform the U.S. political system. For example, is one such voting method that supports the formation of third parties.