Their prospects for gaining entry went from slim to nearly impossible early this month, when the Supreme Court let remain in place a new rule requiring migrants who traveled through other countries to reach the United States to show they have already been denied asylum along the way before they can apply for refuge on the border.

While some have given up, taking free transportation provided by the American government and United Nations back to their homes in Central America, many others who are stuck in Matamoros said that desperation had led them to consider treacherous and potentially life-threatening border crossings — by charging across the river, climbing into hot and airless tractor-trailers driven by human smugglers, or both.

The emerging scenes of furtive crossings reflect an important change in dynamics on the border. Over the last 18 months, many of the hundreds of thousands of migrants who have crossed have sought out Border Patrol agents, rather than hiding from them, in order to request asylum.

The Trump administration’s new policies have played a role in sharply reducing these numbers, but they also appear to be driving people further underground. In a single day last week, Border Patrol agents in the city of Laredo, Tex., apprehended 111 people crammed into the backs of three different tractor-trailers, one of which was 104 degrees inside when it was opened.

A caller alerted the agents to one of the vehicles, sending more than a dozen officers from the local police and the Border Patrol to the scene. The agents surrounded the unmarked truck, and prepared to grab anyone who tried to flee. When they unlocked the trailer, they found 41 men and women, plus two 17-year-olds, seated quietly with their backs against the wall. Some had tears streaming down their faces as they jumped to the ground and were loaded into police vans.