It's best if you have an early flight to find a company that calls Southampton home. I have found four, Totally Driven UK, Twelve Transfers, West Quay Cars and Radio Taxis. There may be more. The only one I am familiar with is West Quay.
International Friends and other companies out of London share drivers if that matters to you.
We are looking for this summer and rates have really gone up since we did this a couple of years ago. One way to control costs is to share the ride with other passengers on the cruise.