One year into his first term in Congress, Senator Eric Schmitt, Republican of Missouri, has sought to find his lane while learning how multilayered relationships in Washington can be.

Mr. Schmitt, a towering figure at 6-foot-6, is a hard-right conservative and staunch defender of former President Donald J. Trump. He introduced 11 bills his first year in Congress, including bills to cut diversity and inclusion offices across federal agencies and to require agencies to roll back three pieces of regulation for each new one. As the Missouri attorney general, Mr. Schmitt signed on to a lawsuit seeking to overturn the results of the 2020 election, and he filed suits against China over the coronavirus and against school districts for their Covid-19 mask mandates.

Even as he has connected with his Senate peers on the right, however, Mr. Schmitt has also forged a deeper kinship with an unlikely colleague: Senator Maggie Hassan, Democrat of New Hampshire.

They have little in common in terms of politics or legislative priorities. But both have children with disabilities: Ms. Hassan’s son, Ben, 35, has severe cerebral palsy. Mr. Schmitt’s son, Stephen, 19, is nonverbal and has tuberous sclerosis, epilepsy and autism.