Should I take “No” for an answer?

Don't Take 'NO' for an Answer - Keep the Door Open
Consider all the people that you construction interview with as networking contacts. Follow-up with a construction personal note keeping them informed of your continued interest in their construction employer and the progress of your job search. When you land be sure to drop them a note and let them know where your job search has taken you. Maintain contact and you may well become a construction executive in the future.