I think that it was mentioned earlier in this thread that the traveler has a free choice of inertial frames.
I suppose, but not if the traveler wants his position to remain fixed in his frame and his clock time to be continuous in his frame. We are not picking some random choice of the new inertial frame at turn around, we are choosing the unique inertial frame that is consistent with the traveler's local frame (which is continuous).
Real on-board clocks will not "jump", and consequently the home clock will also not jump to a later time if the traveler passively uses an extension of his on-board clocks to determine time at home.
No, the traveler's clock never jumps in the traveler's frame, but the home clock does jump in the traveler's frame. Nothing is magical here, the traveler's frame abruptly changes its plane of simultaneity from the outbound IRF to the inbound IRF. So we are forced to admit that the simultaneous reading on the home clock "jumps" in the travelers frame.
Instead, the traveler will first have to synchronize the on-board clocks to the new rest frame. Thus the simultaneous reading on the home clock is adjusted by the traveler, according to his/her choice of instruments and "maps".
No, I intend to do it the other way around, the on-board clock defines the current time in the new rest frame. There is no free choice for time in the new rest frame because we have constrained it to be consistent with the traveler's non-inertial frame. The requirements for the traveler's frame are:
1) Orientation in space is fixed (the traveler is not rotating).
2) The traveler's position in the frame is fixed. (He is at rest in his own frame).
3) The traveler's clock is continuous (it can't jump in it's own frame, his clock is a good clock).
Finally just to be absolutely clear the "jump" in the home clock is not physical; it is rather a "jump" in the perspective in the traveler's frame. I can make the distance stars move faster than the speed of light in my frame just by rotating my body. It is not physical however.