As I said in my comment, the standard things suggested by Jordan are simply wrong for this problem. They are designed to solve very different problems.
No, csape is not appropriate here either. Sorry.
Could you write what is effectively a spline code in multiple dimensions, that uses both the two points, plus the normal to the curve information? Yes, I see no reason why you could not do so in theory. No, I won't write the code for you, nor will it be trivial to write. It simply will not be one of those simple 3 liners.
Finally, it is not at all obvious to me that an interpolant will do better than what came from the green (mid-contour) line, at least not without a fair amount of effort that you will need to invest. Remember that there will ALWAYS be some degree of error from any interpolation. It is not at all obvious that the normals shown would give better information, since I'll bet there are other factors that are just as important. Wind for example, engine speed, slippage, etc. A plane will not always go in the direction it is pointed.