Try Palatka City Park: 40 miles more or less upriver from Jax, lazy river current at ramp, nice courtesy ramp and dock, OK with City to leave tow vehicle and trailer overnight in large lot, good depths in river and approach to ramp, river still wide enough to tack if wind is on nose. There are some ramps closer to Jax operated by Duval County and or DNR, but the closer you get to Doctor's Inlet and further down river the more the tide can affect your ability to launch/recover from a trailer.
Steve