PORTLAND, Maine—Hiring manager Cindy Caplice didn’t rely on job postings to fill 100-plus vacancies over the past year at Sigco Inc., a metal and glass fabricator outside Maine’s largest city. Instead, she took a recruiting outing to the Portland Adult Education center, where refugees and asylum-seekers from several African nations polish their English.

“We cannot grow our company without the immigrant population. We simply can’t do it,” said Ms. Caplice, who now bustles around the 130,000-square-foot factory waving to hard-hatted...