The book Dollars over Dominion: United States Economic Interests in Mexico 1861-1867 is exactly the book you want to answer your question.

In short, Mexico was coming out of the War of Reform (a conflict between Mexico's liberal and conservative factions that ended with Benito Juárez as the president) at the start of the civil war. American-Mexican relations had been strained by the Mexican War, but they became more willing trade partners after Maximilliano was installed as Emperor by the French in 1864 till his deposition and execution by Juarez in 1867.

Anything more specific than this would be speculation. But, from the book, here are the table of contents:

I An End to Forty Years of Enmity in Mexican -United States Relations

II The Contest for Position: Corwin and Pickett, 1861 -1862

III Treaty Making: Attempts at Building a United States-Mexican Friendship, 1861-1862

IV Confederate Relations with Two Mexicos, 1862 -1865

V United States Domestic Politics and the Deterioration of Relations with Mexico, 1862-1865

VI The Tangle of Two Wars: French Intervention in Mexico During the United States Civil War, 1861-1865

VII Untangling Two Wars: French Intervention During the Early Reconstruction Years, 1865 -1867

VIII Mexico and United States Postwar Politics

IX Dollars over Dominion: The Developing United States Economic Interests in Mexico, 1861 -1867 Epilogue Bibliography Indexpage 311

For post war stuff, check out Confederate Migration to Mexico (You can read that with a free JSTOR account). See also the New Virginia Colony. Unfortunately for the colonists, after Maximilliano was shot they were mostly killed or left.

This research also uncovered an interesting ethnic group in Brazil: the Confederados

I'm burrying this here, but holy shit is it interesting. The Confederados have their own society, the Fraternidade Descendência Americana and they hold an annual festival.

Confederate flags and southern dresses in Brazil? Confederate flags and southern dresses in Brazil.

Cross post the above to TIL or pics for karma at your leisure.

Another question, to flesh out certain aspects :How was trading with Mexico different during the war was the south trading more with Mexico from sheer proximity? What was the south's/the U.S's view and relation on Mexico after the war? (how did it contrast with before?)