Senior Democrats, thinking about both the fall election and a potential re-election in four years, said the right choice would stir up enthusiasm on the campaign trail, enhancing Mrs. Clinton’s strengths while not outshining her or overtaking events.

Mrs. Clinton is warm and personable one on one, the Democrats said, which creates an easy camaraderie when she teams up with people she likes.

In San Antonio, Mrs. Clinton seemed to light up in the presence of Julián Castro, the city’s former mayor and now the secretary of housing in the Obama administration. She grabbed his hand and thrust it skyward at an outdoor rally, and the two glided naturally on the rope line, snapping selfies.

The photos from the event, with Mrs. Clinton, 68, beaming next to Mr. Castro, 41, turned out so well that cable networks often show the campaign’s images of the two together on screen when Mrs. Clinton calls in for phone interviews.

On a Saturday night in Youngstown, Ohio, Representative Tim Ryan and Mrs. Clinton made a surprise visit to O’Donold’s Irish Pub and Grill for a couple pints of Guinness. Mrs. Clinton leaned comfortably on the bar next to Mr. Ryan and smiled widely as “Born in the U.S.A.” blared from the jukebox.

Mr. Ryan, holding his pint in the air, as the packed bar followed suit, said, “To Hillary! The next president of the United States.” Mrs. Clinton threw her head back and laughed, then took a long swig of beer.