According to the sectional appendix, Basingstoke up branch sidings (North Yard) are accessed off the Reading lines, it is where NR store tampers etc.
At the London end of Basingstoke on the main lines the up sidings are named Barton Mill Carriage Sidings, and the down side 'Long Valley Sidings'; however SWT refer to the latter as 'Down East End Siding'.
At the country end they are just down sidings and up sidings in the sectional appendix. The former is referred to by SWT as 'Basingstoke Down Yard', and the latter was electrified a couple of years ago, but I've never seen it used by SWT, who call it 'Basingstoke Up Siding West'. Perhaps it sees trains overnight?
Unfortunately that doesn't actually answer where they will expand at Basingstoke, but there are loads of possibilities there...
At Fratton, wouldn't it be possible to just electrify various sidings and loops parallel to the down line and the fuelling point, and enter and exit to the north? They could theoretically bring some more early trains into service at the likes of Hilsea, Cosham or Havant, they don't necessarily have to run south to enter service.
Such a widespread increase in stabling facilities is likely to force a number of changes into early and late services, or increase ECS mileage considerably.