Mr. Schumer, poised to be the incoming Senate Democratic leader, and the current leader, Senator Harry Reid of Nevada, met with Mrs. Clinton’s top campaign aides in Washington last month and pressed them to offer financial support for the Senate races, according to a Democratic official briefed on the meeting. And Mr. Schumer has not been shy since about his hope that if Mrs. Clinton clearly appeared on her way to winning the race, she would redirect some money to congressional races.

“This is one of many things that the Clinton campaign is doing to help us win a majority in the Senate,” Mr. Schumer said through a spokesman.

While party strategists are glad to have the money that Mrs. Clinton is directing from the Democratic National Committee to voter-turnout efforts in Indiana and Missouri, they have little appetite for Mrs. Clinton to visit those states, where she is likely to lose, because that would make it easier for Republicans to tie Democratic Senate candidates to her.

Mrs. Clinton is also pouring money into two congressional districts, in Nebraska and Maine, that both apportion their own presidential electoral vote and have competitive House races. And she is sending an additional $6 million to seven presidential battleground states with hotly contested Senate and House campaigns.

Democrats are also attempting to unseat Senator John McCain of Arizona from the seat he was first elected to in 1986, but Mrs. Clinton’s late decision to swoop into that state is not related to his race, which few Democratic leaders believe they can win. Her incursion there is about her own campaign — and the Democrats’ desire to focus attention on the damage Mr. Trump has done to Republicans with Hispanics.

In particular, Democrats hope to make an example of Sheriff Joe Arpaio, an ardent Trump supporter, by defeating the Phoenix lawman, whose incendiary comments about Hispanics and aggressive tactics with immigrants have garnered attention far beyond his jurisdiction in Maricopa County.

“If Democrats were going to win in Arizona in 2016, you’d need a Republican who turns off Republican women, who really energizes Latinos, and you’d need other races on the ground that can really drive engagement — and we have all that,” said Andrei Cherny, a former state Democratic chairman.