When Mr. Romney ran in 2012, his strategists cast him as a repairman for the country’s floundering economy. But many voters never warmed up to him or came to believe that he cared about them.

By contrast, Mr. Bush frequently talks on the trail about the need to “show what’s in my heart,” aiming to persuade voters that he has their interests and their families’ concerns in mind.

“With Mitt, there was a strategic assumption that the election would be a referendum on President Obama,” said David Kochel, a top Bush campaign strategist who previously worked for Mr. Romney. “Now there’s a recognition that you have to have a little bit of an open window into who the candidate is.”

The documentary “Mitt,” released after Mr. Romney lost to Mr. Obama, captured a man his friends and family knew — a fun-loving husband and father, loyal church servant and turnaround expert.

Mr. Bush’s aides, some of whom believed Mr. Romney would have benefited if that documentary had come out before the election, assigned a videographer to capture the quirky — dorky but affable — Mr. Bush in videos posted frequently on YouTube from the campaign trail, like one in which he describes stumbling upon “Sharknado 3” late one night in his hotel room.