URBANDALE, Iowa — Senator Ted Cruz’s leading Iowa supporters say his get-out-the-vote operation is the best they have seen for a presidential campaign here. He had better hope they are right.

With his monthlong lead in the polls erased, Mr. Cruz’s hopes for pulling out a much-needed victory over Donald J. Trump in the Iowa caucuses on Monday now rest in the hands of thousands of campaign workers and supporters who are spending this weekend telephoning, emailing and knocking on the doors of likely caucusgoers.

Mr. Cruz’s campaign boasts a chairman or chairwoman for each of Iowa’s 99 counties, captains in 1,537 of the state’s 1,681 precincts, and 10,000 people from this state and beyond who have volunteered to help in the final push.

But it is precisely whom all those Cruz backers are striving to reach, and reach again, that illustrates the differing approaches and expectations of the Cruz and Trump campaigns.