How would you find an escape orbit which includes a given starting point (not assuming a circular starting orbit) and escapes the sphere of influence of the current body with a given remaining velocity?
A problem perhaps similar to Lambert's Problem, but instead of an arrival location and time, it is a final velocity.
I have a solution that someone suggested to me, that takes the final velocity as a hyperbolic excess velocity, uses that to calculate specific energy and semi major axis of the departure orbit. However they then assume that you're departure burn will occur at the periapsis of the new orbit and derive the eccentricity, and determine other elements of the departure orbit using that. The approach would work, though you have to use various guesses as the starting position until you find one that results in an orbit departing in the correct direction and the start is always at the periapsis of the departure orbit.
I am hoping that there is an answer to this problem that doesn't require starting at the periapsis of the escape orbit, as I think that would broaden the range of potential departure directions from a given point. However I don't know how I would begin to calculate that.