Minnesota Sen. Amy Klobuchar is quietly gaining momentum in the race for president as rivals drop out

With the smell of manure hanging outside in the cold air, Amy Klobuchar walked into Timbukbrü, a bar so remote that its slogan reads, “Welcome to the other middle of nowhere.”

After greeting the 20 local Democrats waiting inside, the Minnesota senator started sipping a beer, cracking open peanut shells and ticking through several top issues in her campaign for president. After explaining she favored improving Obamacare over passing “Medicare For All,” McKinley Bailey nodded in approval.

"That’s why you’re in my top three,” Bailey, a 39-year-old former Iowa lawmaker,...