Jenny is a sweet name. I think it's fine on its own.
Full names for
Jenny:
JennaJenaeJeanette /
JeannetteGenevaGenevra - I think it makes more sense to use Genny, but
Jenny works.
GenovevaGenistaImogenIphigenia /
Iphigeneia
Perhaps
Ginny appeals to you as well? I like it on its own or as a nickname for
Virginia.