By adding a third, overlapping political influence, I feel that it would necessarily establish a more contiguous legislative and even judicial trend.
I don't know why people keep bringing this up, as if it would even work.
The problem with the American system is that, because for the most part it's winner-take-all first-past-the-post percentage based, any third party would mean the party most opposite to them would always win. If you have 51% of the vote one time, and 49% of the vote the next time, and it keeps switching back and forth every election, then what happens if a party that's very similar to you (like Democrats compared to Green party) gets 5-10% of the vote? Or worse, 25+%?
Then that makes you got between 25 to 45% of the vote, the republicans can get maybe 45-49% of the vote (I'm sure some people would go from republican to green, but their bases aren't exactly 100% overlapping, yeah?) and then the Republicans win. And the next time, greens take from democrat party, republicans get under 50%, and win again. And again. And again.
Unless and until the American system starts switching over to proportional, where (for example) the House of Representatives are based on percentages in the entire state rather than who gets the plurality in a single district, then this will keep happening. Eventually, as people get fed up with never having a say in government and pivot so that everyone votes on the same party instead of splitting the vote, the system will re-stabilize on either the Democrat/Republican two-party system again, or Green-Republican two-party system again, but either way, the third party will wither and die back to the point where it no longer acts as a spoiler to their closer ideological ally.
tl;dr it is literally impossible for a third party to become viable in America, unless and until structural changes to the voting system are made, and saying "man it would be great if we added a third party" is completely ignoring this