Iowa State Senator Brad Zaun (R-Urbandale) has pre-introduced a bill to provide that when no one gets as much as 35% of the vote in a partisan primary, the government will hold a run-off primary. Iowa primaries are in June, so presumably a run-off primary would be in late July or early August.
Under current law, when no one gets as much as 35%, the party then chooses someone by convention. Zaun has been planning this bill for almost six months. See this story from July 2014 which explains why Zaun doesn’t like the current convention system. He ran in the Republican primary in 2014 for U.S. House, and placed first in the primary, but received less than 35%. At the convention two weeks later, he was defeated by the candidate who had placed fifth in the primary. That candidate, David Young, when then elected to the U.S. House in November. Zaun was in the middle of his four-year term as a State Senator in 2014, which is why he is still in the State Senate. All states that now use run-off primaries are in the South, and the number of states with run-off primaries has been gradually diminishing over the last fifty years. Thanks to Nathan Hetzel for this news.