I don't know if JL ever went to MAS's house in Kenner or not but like you point out, she'd need a car to get there.
As far as spending time with her family goes, she'd have to have a way to get there too. Her brother said that she called him to come pick her up and, I believe, it was a distance. So, on her time off while in New Orleans, she may have gone various other places because going 'home' wasn't an option.
That said, just because somebody says something doesn't make it true. There would have to be witnesses and other ways to back up claims that a relationship outside the workplace existed between JL and MAS.