One of the many items I like about the book “The Death of a President” by William Manchester is how he expands on some of the conceptions that others tend to just touch on. An example is the matter of jurisdiction for the investigation of the assassination. Most texts just tell us that there wasn’t a federal statute to cover the murder of a president, so it was a local jurisdiction instead of a federal one. Here is another snip from the above referenced book:
Since 1902 every Secret Service chief had urged Congress to outlaw Presidential assassination, and all had failed. Threatening the life of a Chief Executive was illegal, but if the threat were carried out, if the bravo succeeded, the U.S. Code was silent. There was one exception. Should the assassin be part of a plot, the FBI could move in. This assassin had acted alone, however, and as soon as that became clear local authorities had exclusive jurisdiction. He was guilty only of a Texas felony. Technically, there was no difference between the shooting of the President and a knifing in a Dallas barroom.
I underlined the exception involving a plot. So, it seems to me that the FBI would have been looking for any possibilities of a plot from the very beginning. Shortly after LHO’s death, LBJ ordered the FBI to investigate. But before that, I believe the FBI had a strong incentive to look for a plot (aka: conspiracy).