There are several real estate offices that also offer rentals, along with individuals like my self that have rentals for lease.
You will find a few "large" apartment complexes in JC as well and many "small" apartment buildings.
if you have a pet you should state that first because a lot of rentals to do not allow pets. State how many bedrooms, bathrooms, etc. you are looking for and your price range.
2 bedrooms will go for $600 and up; 3 bedrooms $800 and up and 4 bedrooms $1,000 and up.
I wish you well in your search.