According to Anthony Summers 1980 book "Conspiracy" Nina Garner says the FBI called to her 3 weeks after LHO arrived in New Orleans:
With Oswald heading to New Orleans on April 24th 1963, this would have been in mid-May 1963. (Oswald actually moved in to 4905 magazine street on May 10th 1963)
Summers repeats the story in his 2013 version of that book which was then retitled "Not In Your Lifetime". The 2013 version leaves out the mention of LHO being under surveillance from a car, probably because this was second hand information i'm guessing Nina Garner had rather than actually seeing this herself.
The question then is how come Nina Garner never spoke about being questioned by the FBI in May 1963 in either her WC testimony or the Clay Shaw trial testimony?:
WC Testimony:
https://www.jfk-assassination.net/russ/testimony/garner_m.htmClay Shaw trial testimony:
https://www.jfk-assassination.net/russ/testimony/garner_sh.htm