Assuming:
- non-spinning SOIs - they only undergo translational motion along the planet's orbit, but direction of their axes relative to each other is always constant. That's the trivial approach; sun-synchronous orbits are impossible and tidally locked bodies will need spin equal to orbital period, instead of a flat 0, but calculations are made easier.
- The "root" frame of reference is the Sun's SOI (Sun is assumed to be immobile).
- We're not concerned about the motion of the craft, just motion of the conics.
The hyperbola exists in Earth's SOI and is stationary relative to it (speed relative to Earth SOI = 0); its endpoint is its part and so, the endpoint's velocity in the "root" frame of reference will be equal to speed of Earth's SOI at that point - that is, Earth velocity.
This would become more complex in case of spinning SOIs, requiring to trace the velocity of the SOI edge relative to Sun, but as I understand this is not our case.