You might think the Panama Papers leak would cause the ultrarich to seek more transparent tax havens.

Not so, according to Jordan Greenaway, a consultant based in London who caters to the ultrawealthy.

Instead, they are going further underground, seeking walled-up havens such as the Marshall Islands, Lebanon, and Antigua, Greenaway, who works for the PR agency Right Angles, told Business Insider.

The Panama Papers leak around Mossack Fonseca, a law firm that helped politicians and businesspeople hide their money, has increased anxiety among the rich over being exposed, Greenaway told New York reporters in a meeting last week.

"The Panama Papers sent them to the ground," he said.

As of 2012, the world's richest people held as much as $32 trillion in offshore tax havens, according to research by the Tax Justice Network.

Meanwhile, there's a general concern among professionals that more enforcement is to come. Deloitte, a consultant, found in a survey that most professionals expect increased global enforcement of financial crime after the Panama Papers.

Part of the shift to more secretive havens is a result of other tax havens becoming more transparent, which the rich aren't happy about, Greenaway said.

For instance, Guernsey, an island tax haven off the coast of France, said earlier this year that it supports moves to improve transparency following the Panama Papers fallout. Other tax havens haven't caught on, meaning they are attracting those who would rather stay secretive, Greenaway said.

Other factors are also at play. Financial advisers to the rich have an inherent interest in putting their clients' money in legally complicated structures, he said.

Emotions also play a role.

"That anxiety is even more intense with inherited wealth," Greenaway told Business Insider. "The whole concept of having to provide details on their finances is alien to them."

"It will take more scandals to shift their behavior," he added.

Still, recent leaks have caused the rich to take more cautious steps. Emails, for instance, are becoming less common, said Paul Blanchard, Greenaway's colleague who runs Right Angles. They're talking deals over the phone rather than in writing, he said.

"There's a Luddite move," Blanchard said.