A HSL project of this sort is probably going to be in the 'do not go here' cupboard for ministers for a generation, given the generally negative press HS2 has attracted.
HS2 to Birmingham being a major success when it opens, either directly or as a result of benefits to the southern WCML corridor may change the perception, but those benefits will take some time to become apparent.
Tactically, the way to approach this in future maybe to pack up a series of incremental projects that add up to a new HSL corridor. A HSL from Hansacre to Crewe would have to be up there to ensure HS2 fully delivers as a southern WCML relief line.
After that, connectivity between West and East Mids would also be on my agenda, either a de-scoped HS2 phase 2 (East) or revisiting the Birmingham to East Mids enhancements considered in the HS2 Strategic Alternatives Final Report. Alternatively or as well, I'd want to implement some of the other upgrades to the ECML corridor also outlined in the HS2 Strategic Alternatives Final Report, including a section on new line from Alexandra Palace to Biggleswade. Dusting down some of Virgin's ECML bid from the early 2000's, which included a section of new line from Peterborough to York is also something I'd look at.
I'd be aiming to ensure all three north south mainlines are relieved but also to ensure the core of the cross country network also benefitted.