By the way, Trotsky did attempt to get into the US in the late 1930's. For one thing, he had not lost his interest in America; for another, he thought he would be safer in the US than in Mexico. (President Cardenas was staunch in resisting Stalinist pressure to expel Trotsky, but Cardenas' term would end in 1940, and he was not eligible for re-election. And by 1938 Trotsky was already convinced that Stalin's agents were staking out the Trotsky residence near Mexico City in preparation for an assassination.) All his attempts were in vain. In an interview in December 1938 he stated his frustration, noting that twenty-two years earlier he had gone to New York "without any passport, without any visa, without any absurd and humiliating formalities! Your immigration authorities were interested that I did not have trachoma but were absolutely unconcerned about my ideas. Yet twenty-two years ago, my ideas, I venture to assure you, were as bad as now..." https://www.marxists.org/archive/trotsky/1938/12/nynews.htm
It is in this context, incidentally, that Trotsky's much-criticized willingness to testify to the Dies ("Un-American Activities") Committee (presumably about Stalinist activities in Mexico) must be considered. The Committee's request for him to testify meant that at last he had a chance to get into the US--originally he might just have a six-month visa, but even that would buy him time, and perhaps it could be extended...
This actually makes a more interesting POD, at least IMHO.