Republican senators are confronting a political challenge that is increasingly hard to ignore as they engage with voters during the July Fourth recess: Under their health-care overhaul, average premiums for a midlevel insurance plan would jump by 20% next January.

That means many people who don’t get insurance through work would see their premiums increase just a few months before the midterm elections, according to the nonpartisan Congressional Budget Office. Premiums would fall in later years, in part because less-comprehensive...