WASHINGTON—When Doug Jones arrived in the Senate as a Democrat from deep-red Alabama, he could have sought to boost his re-election chances by siding with President Trump on high-profile issues. For the most part, he hasn’t.

In his short tenure in the Senate, Mr. Jones has pushed Alabama to expand Medicaid and has criticized the president for attacking the Affordable Care Act. He voted against the president’s Supreme Court nominee Brett Kavanaugh. But Mr. Jones hasn’t been a party-line Democrat: He has voted for the majority...