You might want to read Accessories After the Fact. The author did a great job of documenting Oswald doing some kind of low level undercover work.
What I always find amazing is people who buy into the official story ALWAYS seem to have plausibility amnesia when it comes to this case. For example, they always make it as if Oswald was some lone wolf type of guy like Tim McVeigh who just decided one day to blow Kennedy's head off.
He wasn't but don't take my word for it - read Simpich's State Secret. When Oswald said he was nothing but a patsy and explained that his head was pasted onto another's body in the BYP's, the truth is right there in front of you.