In my experience, jojoba oil does not go rancid on the head, even when left there for weeks at a time. So you’ll be just fine!
However, jojoba oil cannot and does not make hair grow. Products—including oils—don’t really have anything to do with hair growth. Hair growth is primarily a hormonal process, so topical treatments tend not to do much.
The hormones involved with hair growth can change with age, diet, environment, and stress levels.
For tips about how to combat thinning hair, check out this post. Let us know if you have any other questions; we’d love to help!