As the final ballots are tallied in the still-uncalled races across the nation, Democrats have a net gain of 32 House seats and have a chance at picking up as many as 38, according to unofficial counts.

The Associated Press has yet to call the winners in 10 closely watched House races, which are expected to split evenly between the two parties. Democrats have already flipped 35 Republican districts, and GOP candidates captured three Democratic seats.

The Democrats picked up another Senate seat Monday evening when the Associated Press called the Arizona race for Kyrsten Sinema, who defeated Republican Martha McSally by more than 38,000 votes, or about 1.7 percentage points. Ms. Sinema, the state’s first Democrat in 30 years to win election to the Senate, will succeed retiring Republican Jeff Flake.

With Ms. Sinema’s victory, the Democrats have now flipped two Senate seats and the Republicans three, with the Florida Senate race still outstanding.

As the ballot counting continues, Congress and the 2019 class of freshman will gather Tuesday in Washington to close out the current session and prepare for the next.

“The early election-night analysis overestimated the size of the Republican gains in the Senate and underestimated the size of the Democratic gains in the House,” said Nathan Gonzales, the editor of the nonpartisan Inside Elections.

Some of the races outstanding have been overlooked as both the Senate and gubernatorial races in Florida are in the throes of a recount process that has spurred a flurry of legal activity.

A runoff election on Nov. 27 will determine if Sen. Cindy Hyde-Smith (R., Miss.) retains her seat against Democrat Mike Espy. Ms. Hyde-Smith faced national criticism this weekend after a video posted to Twitter featured her saying she “would be in the front row” of a public hanging if a supporter invited her. The comment was quickly condemned as an allusion to lynching, and Mr. Espy, who is black, called it “reprehensible.” Ms. Hyde-Smith said the comment was an “exaggerated expression of regard, and any attempt to turn this into a negative connotation is ridiculous.”

Three of the uncalled House races are in California, where it may still be weeks before election officials finalize their results. In the California 10th District, Democrat Josh Harder has taken the lead against incumbent Republican Rep. Jeff Denham. Republicans Young Kim and Mimi Walters have held on to slim leads in the California 39th and 45th, respectively, though their leads have dipped as more votes have come in. The AP has already called two close California races to Democrats after election night.

“In the California 39 and the California 45, where Republicans are leading, it just keeps getting closer,” Mr. Gonzales said. Overall Democratic gains could be “in the neighborhood in 38, 39, but some of these races are becoming so close, and I’m not sure where they’re going to end up,” he added.

Republicans enjoy slight leads in three of the other uncalled races, in the Texas 23rd, Georgia 7th and the New York 27th congressional districts.

Rep. Will Hurd (R., Texas), Rep. Rob Woodall (R., Ga.) and Rep. Chris Collins (R., N.Y.), who was indicted on insider-trading charges this year, are on track to return to Congress. Mr. Woodall’s Democratic opponent, Carolyn Bourdeaux, has taken legal action challenging how election officials have counted absentee ballots.

Gina Ortiz Jones, the Democrat who challenged Mr. Hurd, appeared about 1,000 votes short of ousting the two-term incumbent. Ms. Ortiz Jones’ campaign is holding out for provisional ballots and those sent in from military members abroad, one of her aides said. Her campaign hasn’t ruled out requesting a recount once all votes have been counted.

Mr. Hurd has already declared victory, but the Associated Press hasn’t called a winner in the contest.

“We’ll have a better idea at the end of the next week,” Ms. Ortiz Jones said in a telephone interview.

In the Utah 4th, New York 22nd and New Jersey 3rd, Democrats appear likely to flip more Republican-held seats.

And a contested race in Maine’s Second District has proceeded to an unusual ranked-choice recount, when election officials will count the second and third choices of voters who supported independent candidates not in the top two. That contest also leans toward the Democrat Jared Golden over Republican incumbent Rep. Bruce Poliquin.

—Joshua Jamerson contributed to this article.

Write to Andrew Duehren at andrew.duehren@wsj.com