SWR run direct Salisbury-Reading services on a Sunday, so a single change a Salisbury is also possible.
However that would take a lot longer than changing at Basingstoke & it would put someone on the GWR service as the next train to Southampton.
And somehow I doubt a Reading - Southampton ticket will be valid via Salisbury.
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As for the journey you are thinking of making, I have seen SWR advances for Southampton to Reading on weekday evenings with someone from Southampton catching a Weymouth - London service to Winchester, changing to a Portsmouth - London service to Basingstoke before changing to one of the 2 SWR trains to Reading. Sundays it should be one change as both trains from Southampton call at Basingstoke