I wouldn't say that Mexico was uninvolved in the Cold War. To offer but one example, the century old Chamizal dispute between the U.S. and Mexico was settled in the 1960s largely because of Cold War politics and maneuvering.

To offer some context, frequent shifts in the Rio Grande (which was designated as the international boundary between the Gulf of Mexico and El Paso by the Treaty of Guadalupe Hidalgo in 1848) led to the United States claiming several hundred acres of Mexican territory left in El Paso Texas as a result of torrential rains and floods in the 1860s. Though Mexico appealed to the United States for decades and a 1911 arbitration court ruled that the United States should return the majority of the land Mexico, the United States Government continued to claim the land as its own. (This, itself, is a fascinating and complicated story, one that you should investigate if you're interested in border studies and U.S.-Mexico relations in the 19th and 20th centuries). Though U.S. and Mexican officials from 1911 to 1960 discussed the dispute and possible solutions, no deal was brokered until John F. Kennedy's administration.

JFK wanted a deal that would abide by the terms of the 1911 arbitration (or something close to them), returning most of the disputed Chamizal land to Mexico. He wanted to do this, in part, because he believed keeping the Latin American world happy would prevent those countries from supporting Communism. In 1962, he met with Mexican President Adolfo Lopez Mateos in Mexico. The visit was meant to highlight and reinforce friendly relations between the U.S. and Mexico. The leaders agreed to broker a deal on the Chamizal dispute during the meeting as well. U.S. Ambassador to Mexico, Thomas C. Mann, and his Mexican counterpart, along with officials from both countries, negotiated a solution that returned most of the land to Mexico, redrew a section of the international boundary, and redirected a portion of the channel of the Rio Grande. The treaty resolving the century long dispute was signed in August 1963. After JFK was assassinated, officials in the U.S. and Mexico continued to work to make the terms of the treaty a reality. Today, a National Park Service site exists in El Paso, Texas to commemorate the peaceful resolution of this dispute. A sister park is located across the international bridge in Juarez. Again, though not the only reason a deal was reached between both countries, the Cold War served as a major impetus for the U.S. to maintain a good relationship with Mexico.

There are several great articles on the Chamizal dispute and resolution that you can locate if you have access to JSTOR. Two scholarly monographs (one that is rather old and out of print and another recent publication) are also useful. The first, published in 1965, was written by Sheldon Liss and titled "A Century of Disagreement." The other, published in 2018 and written by Thomas Tunstall Allcock, is titled "Thomas C. Mann: President Johnson, the Cold War, and the Restructuring of Latin American Foreign Policy." You can also visit Chamizal National Memorial's National Park Service website where they have a brief, though detailed, history of the dispute and resolution, including a preview of the new permanent exhibit that will open to the public once it is safe to do so.