Southampton Hotels

I am not personally aware of any, however I would be inclined to focus on my hotel and rate then a free shuttle. Most of the hotels in the West Quay Mall area (the new downtown) are about 6 to 10 GBP from the cruise terminals.

....................and in many cases the free "shuttle" is simply the hotel calling a cab & charging the fare to their account.

From edge-of-town or out-of-town hotels that don't provide a shuttle a cab fare will of course be higher, but any cruise terminal is no more than a £10- cab ride from any city centre hotel on this map

https://www.londontoolkit.com/travel/southampton_map.htm

JB :)

There's a brand spanking new Travelodge West Quay that is worth checking out.

For those interested, this hotel has already changed its name to Travelodge Southampton Central Hotel.