I usually see the approach leg before an orbital injection trajectory around a body lying outside Earth's orbit drawn as catching up to it (moving in the same general direction/speed as the body's orbital vector) and approaching the far side of the planet/moon/etc. Does it ever make sense to approach the body from the near side, ahead of it, and still end up in orbit around the body? Can such a trajectory be accomplished without totally blowing your delta-v budget? If feasible and practicable, has any probe taken this approach vector?
In the frame of reference of the Sun, it is the outer solar system body (Mars, Jupiter, etc.) that catches up with the vehicle from Earth, not the other way around. The energy of the vehicle from Earth is lower than the destination body, so that body is going faster around the Sun than the vehicle.
In the frame of reference of the destination body, the approach velocity of the vehicle is a vector fixed by the trajectory from Earth, and defines a plane, called the B-plane, through the body that is perpendicular to that approach velocity vector. The vehicle can choose any point in the B plane to target, which will determine the closest approach distance to the body and a clock angle somewhere around the body. Doing an orbit insertion burn at the closest approach point will result in an orbit that is prograde, retrograde, polar one way or the other, or somewhere in between depending on that clock angle, as you like. Though the orbit is limited to an inclination no less than the declination of the approach vector. So you can't go into an equatorial orbit with one maneuver.