One answer to your question depends on what you mean by "political." The Supreme Court is obviously political in the sense that it is a crucial player in the politics of governance. A more narrow sense of the world political might mean 'ideological.' In this sense, I would argue that since each member of the court undoubtedly has always had private opinions, beliefs, and values shaping their decisions on rule of law, then in this sense, too, I would argue the court is political. However, I think you mean "politicized" as in partisan politics, and in this sense, there is a more specific history at play.
I'm going to recommend two podcasts here that tell two different stories -- each citing/interviewing real historians and using thorough research -- about a moment when the Supreme Court became more partisan and actively political than it had before:
The most important story that leaps out to my mind -- probably because my speacialty is in World War II and I'm visiting the FDR presidential library soon -- is Franklin Delano Roosevelt's poorly planned, not-executed, and politically costly attempt to "pack" the Supreme Court. As most enthusiasts of history know, FDR launched the most intensive Presidential agenda ever, The New Deal, largely in an amazing productive spurt of energy: his first 100 days in office in 1933 are used as a benchmark for other Presidents actions (or lack thereof) to this day. Unfortunately for Roosevelt's agenda to haul the nation out of the Depression through the power of an activist federal government and interventionist state, the Supreme Court began striking down Roosevelt's most famed programs on the grounds that they were unconstitutional, including the National Recovery Act and the Agricultural Adjustment Act.
However, FDR had just won the 1936 election by the greatest landslide in American history, and saw a completely constitutional way to limit the power of the Supreme Court so he could continue his New Deal programs uninterrupted. On February 5, 1937, FDR announced his intent to add up to six members to the Supreme Court, bringing the total number of justices to 15. Immediately, the plan was denounced as a transparent ploy to appoint 6 justices of his own picking to support his own agendas-- "court-packing" as the newspapers denounced it. However, Roosevelt pointed out that the constitution does not actually dictate how many justices should be on the Supreme court; this had only been codified as recently as the Judiciary Act of 1869 -- almost a century after the founding of the country. He also made various arguments for spreading the work of the court more evenly among more members, and replacing older justices with younger, sharper justices through his plan. In one of his famous fireside chats on March 9, 1937, Roosevelt said:
The American people have learned from the depression. For in the last three national elections an overwhelming majority of them voted a mandate that the Congress and the President begin the task of providing that protection - not after long years of debate, but now. The Courts, however, have cast doubts on the ability of the elected Congress to protect us against catastrophe by meeting squarely our modern social and economic conditions. [...] Since the rise of the modern movement for social and economic progress through legislation, the Court has more and more often and more and more boldly asserted a power to veto laws passed by the Congress and State Legislatures [...] The Court has been acting not as a judicial body, but as a policy-making body. [...] We have, therefore, reached the point as a nation where we must take action to save the Constitution from the Court and the Court from itself. [...]
What is my proposal? It is simply this: whenever a Judge or Justice of any Federal Court has reached the age of seventy and does not avail himself of the opportunity to retire on a pension, a new member shall be appointed by the President then in office, with the approval, as required by the Constitution, of the Senate of the United States.
That plan has two chief purposes. By bringing into the judicial system a steady and continuing stream of new and younger blood, I hope, first, to make the administration of all Federal justice speedier and, therefore, less costly; secondly, to bring to the decision of social and economic problems younger men who have had personal experience and contact with modern facts and circumstances under which average men have to live and work. This plan will save our national Constitution from hardening of the judicial arteries.
The number of Judges to be appointed would depend wholly on the decision of present Judges now over seventy, or those who would subsequently reach the age of seventy. If, for instance, any one of the six Justices of the Supreme Court now over the age of seventy should retire as provided under the plan, no additional place would be created. Consequently, although there never can be more than fifteen, there may be only fourteen, or thirteen, or twelve. And there may be only nine.
As you may expect based on the current number of judges on the bench, this plan failed, and failed hard. Roosevelt's bold proposal only thre gasoline on the fire of those who denounced the growth of government under his presidency as proof of executive overreach, painting him as a dictator or socialist wanna-be. The Democratic party itself split over the plan, making Roosevelt look like a weak leader even of his own party. Roosevelt might never have recovered his positive public image if it hadn't been for the start of World War II -- the 1938 mid-term elections went overwhelmingly to anti-New Deal opponents of FDR, and he struggled to do anything against an antagonist congress from then on.
However, ironically, after this kerfluffle, the Supreme Court did begin to uphold Roosevelt's legislation. He had showed his willingness and the constitutionality of legal modes of challenging the Supreme Court, and the court got the message. As Richard G. Menaker writes for the Gilder Lehrman Institute:
On the doctrinal front, the administration had won the war. Beginning with Parrish, the decisions of the Supreme Court upholding government’s power to regulate set the pattern for the balance of Roosevelt’s presidency and for nearly a half century thereafter. The ideology that had stymied the New Deal and parallel state legislative efforts to control private economic relationships went into eclipse. The Court had gone through what some commentators have described as a “constitutional revolution.” Yet the apparent stimulus for turn-about was widely attributed to the court-packing initiative, which had ironically constituted one of Roosevelt’s most embarrassing defeats.
You may also wish to read this Smithsonian Magazine Article about the debacle