If you are on the south side of town check out:
St. John's - Good track, tough.
South Hampton - Good shape, has a good mix of holes and is a lot of fun.
Out to the west in Orange Park, 15-30 mins and easy to get to if you are staying by 95 / Southside Blvd., you have:
Eagle's Landing - I really like this course. There is nothing spectacular about it, but I always have fun when I play there. In good shape, and usually not very crowded. Can be a bit of a drive from Jax, 30 mins.
Eagle Harbor - Ok couses through a housing development. Has a good twilight rate.
Flemming Island - Similar to Eagle Harbor.
Orange Park CC - Only played it once, but remember it having some good holes. Is the closest of the Orange Park courses to Jax. I am not sure about access. I think it is private, but allows visitors at certain times. I had no problem playing by just calling the shop.
Hyde Park is a Ross, but not one of his best. There doesn't seem to be much left of it.
Windsor Park is convenient, but very so so and the conditions are quite hit and miss. Depending on how it is playiing, it can be over priced.
Jax Beach GC is also very convenient. The conditions are questionable and the course is rather boring, but it is close and cheap.