I must give the endpoint of the hyperbola the Earth's velocity first. I can think of several velocities to use, the velocity of the Earth at that moment in its conic orbit around the Sun, or the velocity of a satellite in its own conic orbit around the Sun at the new distance, but there could be other options.
The Earth's SOI is bound to Earth translation-wise, and distant stars rotation-wise. That's what the craft velocity is calculated relative to in the hyperbola trajectory. You don't spin the SOI relative to anything, ever (no daily, no synodic - axis X points towards vernal equinox always.) - only translate it along the orbital path of the planet around the Sun.
That way, linear velocity of every point of the SOI is the same relative to the Sun (and same as the planet's), so regardless of the point of escape you just add the planet's velocity, and the craft velocity, and use that as initial velocity at the same point in the sun's SOI. Same approach on going "down to the planet" - substract the planet's velocity vector from the craft entering, and you have the speed in the planet's SOI.