If a person involved in the investigation is made to feel personally responsible for an error in procedure, or paranoid that the assassin might not get a conviction, then some of the framing and cover-up would be a case of a "naturally forming" conspiracy that doesn't necessarily involve one evil entity bossing everybody around and ordering them to be liars for life. Just think of how paranoid some of the authorities may have been in the following HOURS that their "evil communist" accused assassin didn't leave behind enough evidence for a conviction. Evidence could have been fabricated and substituted without even having to be part of the original homicidal conspiracy.
You could have also described certain actions without assuming that any conspiracy even exists. Just natural human tendencies.
As in a person involved in the investigation actually feels responsible, all on his own, for an error in procedure or a mistake that was made. They could then act in a way that could be interpreted by a rational person as covering their own ass. No conspiracy necessary. Example: Dr. Malcolm Perry.
There could also be instances of comments made that could be interpreted as covering up a conspiracy that could also be interpreted by a rational person as someone acting on there own trying to head off any speculation of conspiracy with good intentions. No conspiracy necessary. Examples: Katzenbach, Hoover
There could also be instances of what looks like a conspiracy to cover up a conspiracy that are actually instances of trying to cover up something else like personal information of JFK's other medical conditions or to keep the presidents body from becoming a spectacle for public display. Examples: Robert Kennedy, Jackie Kennedy.
There are also instances of someone acting in a way that they may initially feel to be heroic. Then later after they hear that there actions actually make them look like a participant in a conspiracy. They then try very hard to clear their name and this is used again out of context to make it look like they are definitely part of a conspiracy. No conspiracy necessary. Example: Jack Ruby
