Someone posted this idea previously (Available From Date for Files), but it was archived, I assume because the title and description were not sufficiently detailed.

When you set a file's availability, you can only specify the dates. The file always displays or is hidden at 12 am on each date (per the attached image).  The date pickers used in Files should be the same as the date picker on most other pages in the system, allowing instructors to specify the date and time of availability.