How do you know that a lot of people fled Mexico and entered the U.S. illegally at that time?
Well, I found an answer to my question. It seems that a great many Mexicans did indeed flee to the U.S. at that time.
However, I found several websites that show that Toribio Romo was adamantly opposed to Mexicans going to the U.S. A distant relative of his (second cousin, once removed) wrote an article for Texas Monthly in 2010, after doing a lot of research about his cousin, Fr. Toribio Romo.
This cousin tells a different side of the story of Fr. Toribio Romo. The Texas Monthly article describes Fr. Toribio Romo's opposition to his parishioners going to the U.S.; and in fact, Father had even written a one-act play, called "Let's Go North!" in which he warned against going north to the U.S.
Here's his description of Fr. Toribio Romo's play:
"If you betray your country and go north, Toribio's play warned its Mexican audience, you might come back as a "rooster hen that neither crows no lays eggs." Or worse, a Protestant. Take a look at what becomes of the Mexican who goes north, Sancho says at the end of the play. He ends up a man without a religion, without a country or home....a coward, an afeminado who is incapable of feeling shame for having abandoned responsibilities to his family. Despite this the roads are packed with Mexicans headed towards the United States, in search of bitter bread. Everywhere you hear the rallying cry - "Let's go north!" "
The author of the article says that Fr. Toribio Romo did everything he could to keep his parishioners in Mexico from leaving home.
https://www.texasmonthly.com/articles/my-tio-the-saint/