Imo Oswald was given a midnight press conference because nobody besides Oswald had anything to hide.
JohnM
We can add that Oswald was allowed to meet with his family - Marina, Robert, Marguerite - and with Lou Nichols, head of the Dallas Bar Association. And make personal phone calls.
He had numerous opportunities - particularly in this press conference - to expose this alleged conspiracy. Or at the very least to give hints/names of something more.
If one believes in a conspiracy involving powerful elements of the government, then the obvious decision for them would have been to allow selected journalists to meet with him. I've read numerous claims about Operation Mockingbird, about how Hugh Aynesworth (and others) were CIA "assets". So, why not allow those "assets" to meet with Oswald, say he's being treated properly, and eliminate any risk of exposure from him? These "assets" would just report what you told them.
Shorter answer: they wouldn't.