Can I changed my ultimate inclination mid-transfer? Where/when?

[THX1138]

I have a craft that is mid route between Pol and Laythe (Tylo helped change my orbit). My intention is to rendezvous with a second craft which is in a roughly polar orbit. What manoeuvre should I make to do this as efficiently as possible? What direction should I thrust and when? I don't know if I can change my final inclination much without losing the encounter and how I should be determining the appropriate direction to burn.

The two images below show the position I am in my transfer and the orbit of the target.

[James Kerman]

Inclination changes are cheapest at the slowest part of your orbit.  Sometimes this is not possible - if so I would capture into a highly elliptical orbit and change inclination at apoapsis.

[THX1138]

10 minutes ago, James Kerman said:

Inclination changes are cheapest at the slowest part of your orbit.  Sometimes this is not possible - if so I would capture into a highly elliptical orbit and change inclination at apoapsis.

but don't I need to be near the ascending or descending nodes in order to get my inclinations to match? Is it optimal to do some of the correction away from the AN/DN and then finish it at the AN/DN? I've always done the inclination changes as close to the AN/DN as possible but using the one furthest from the body I'm orbiting.

Ideally the AN/DN would align with the PE/AP but that doesn't usually happen.

[James Kerman]

I usually intercept inclined bodies at the AN/DN if I'm going into equatorial orbit however for a polar orbit I would do the inclination burn at AP in your diagram incorporated with a retro burn to get my desired PE then circularise at PE.

[OHara]

You can when you are about 90° from your Laythe encounter, during your half-orbit around Jool.  You cannot significantly change the direction from which you enter Laythe's SoI, but you can adjust the position of your periapse with Laythe to be anywhere in a circle roughly perpendicular to your entry direction.

In your picture, you show that your entry direction is very roughly perpendicular to your target orbit, so for this unlucky case you can't help much.  The best would be to mid-course correct so your periapsis is hidden behind Laythe in the close-up view of Lathe that you show, so your periapsis is where you cross your target's orbit.

Describing the same geometry another way, looking at Laythe from the point-of-view of your path entering its SoI, your target orbit is an ellipse; you can adjust your SoI entry location so your path crosses that apparent ellipse at the one of its two vertices (tightest-curvature points) that has your target moving generally away from you.  That gets your angular momentum about Laythe aligned as well as it can be with your target's.

For a circular target orbit like you have, that puts your periapsis with Laythe at your AN relative to your target, so you can capture-burn to leave a high apoapsis (near the DN) and correct inclination while moving slow-ish at the DN.  Looks like the inclination will still be 60° but I don't see how to do better give where you are approaching from.