Union Hospital Of Cecil County in Elkton, Maryland . Health (9 days ago) Union Hospital Of Cecil County is a General Acute Care Hospital in Elkton, Maryland.The NPI Number for Union Hospital Of Cecil County is 1609961390. The current location address for Union Hospital Of Cecil County is 123 Singerly Ave, , Elkton, Maryland and the contact number is 410-392-2686 and fax number is 410-398-0073..