Nope, we will pick upand drop off the rental at JAX.
Alamo shows almost $1200 for those dates at JAX. I doubt picking up the next day will save you any because usually anything more than 5 days counts for another week, so you will be at two weeks regardless of whether you rent for 13 or 14 days.
Mousesavers.com has a lot of good info on rental car rates. Another helpful site is carrentalsavers.com.
The good thing with cars is you can reserve/cancel/rebook as much as you want with no penalty. But that means you end up having to watch things pretty closely to get the best rates when they appear.