According to the Artscroll biography of him by Rabbi Shimon Finkelman, it would be fair to say that Ohio marked purely a "convenient stepping stone" into America.
On pp. 72-73 Rabbi Finkelman notes how his relative Rabbi Nechemiah Kastonowitz who shortened his name to Katz had already left Russia and become Rabbi in Toledo, Ohio. Rabbi Katz made a point of approaching his own congressman who arranged an appointment with Senator Buckley of Rabbi Katz's state of Ohio, which also eventually led to him gaining access to Senator William Edgar Borah, who had served as chairman of the Senate Foreign Relations Committee.
On p.79 the narrative picks up on when the visa was finally granted. It writes as follows:
The visa to America was finally granted, in large measure thanks to a plan devised by Rabbi Nechemiah Katz and approved by his congregation. Rabbi Katz called a meeting of his shul's board of directors and proposed the following suggestion: Should Reb Moshe be granted entry into the U.S., Rabbi Katz would resign his position and it would be given to Reb Moshe. The proposal was accepted. In fact, after arriving in America, Reb Moshe served as Rav of Bnei Yaakov for a few weeks. He then resigned the position and it was returned to his brother-in-law.
Rabbi Katz informed the secretary of the Aguda HaRabbanim, Rabbi Yehudah Leib Seltzer, of his shul's decision and Rabbi Seltzer then contacted Senator Robert Wagner of New York. A letter from the American consul in Riga to Senator Wagner dated November 2, 1936, begins:
I have the honor to acknowledge the receipt of your letter dated October 6, 1936, in which you request to be advised with regard to the status of the application for an immigration visa of Rabbi Moses Feinstein, now a resident of Riga, who desires to proceed to the United States to join the Congregation Bnei Jacob of Toledo, Ohio...
However, even this move did not produce immediate results. Again Rabbi Katz thought of a plan. he appealed to the secretary of the Ohio Democratic Party, a gentile who was know as a warm, sympathetic person. The secretary enlisted the aid of the state's lieutenant governor, who beamed as he listened to Rabbi Katz's story. The American consul in Riga, Latvia, was an old friend of his! He got word to the consul, who agreed to process the papers for the family. So it was that - as Rabbi Katz put it - they had the indescribable privilege of saving the future gadol hador!