Leapy, I will leave the areas to the East of Southampton to those for whom it is home ground. All the following have good sized car parks and i have used
Starting at Southampton, I can recommend the slips at Hythe and Calshot, neither needs the trailer to be unhitched. Calshot is particularly good at any state of the tide, nice and wide and immediate access to the Solent area. Cost £14
Moving Westwards, the next would be Lymington, good slip and pontoons for easy access to boat. Great for access to Western Solent and Needles area. About £15 + parking.
Christchurch next, Mudeford quay is trickey and only usable toward high tide, very busy in summer. Would rather use the Slip at Christchurch Quay, if you use an hour before or after low tide its easy. About £10 for parking.
Poole has a couple of slips, the most popular in Baiter Park, which can get busy with Jetski's and as its a very gentle gradiant, you have to unhitch and hand launch/recover. Very good access to the harbour and rivers, as well as the beaches and Swanage/Jurassic coast.
Finally, Weymouth, very good slip in the Harbour at all tides, parking for trailer, but not car, which has to go in another carpark. Can get busy. God for doing Jurassic coast and Portland. About £13 for launch and trailer. Look at www.boatlaunch.co.uk
hope this helps Steve