They are a bit of a stretch really.
Jenny can be used as a nickname for
Jane and has been for years past but I don't really see where
June or
Jane could be nicknames for
Jennifer, their origins are so completely different. Now if you wanted to use
Jena or even
Veva as they are forms of
Guinevere and therefore
Jennifer, I think that would be fine.
Jinny (by way of
Jenny) would be fine as a nn. Or what about
Effie, or
Enna, or Enner? Fin or
Finn would even be possible because of all the letters being there in
Jennifer, but there aren't A or U in
Jennifer at all. :(