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A well-established way to get more energy from a gravity assist is to fire a rocket engine at periapsis where a spacecraft is at its maximum velocity.
Rocket engines produce the same acceleration regardless of their initial velocity. A rocket acting on a fixed object, as in a static firing, does no useful work at all; the rocket's stored energy is entirely expended on its propellant. However, when the rocket and its payload are free to move, the force applied by the rocket during any time interval acts through the distance the rocket and payload move during that time. Force acting through a distance is the definition of mechanical energy or work. Ergo, the farther the rocket and payload move during any given interval, (i.e., the faster they move), the greater the kinetic energy imparted to the payload by the rocket. (This is why rockets are seldom used on slow-moving vehicles; they were simply too inefficient when used in that manner.)
Energy is still conserved, however. The additional energy imparted to the payload is exactly matched by a decrease in energy imparted to the propellant being expelled behind the rocket. This is because the velocity of the rocket is being subtracted from the propellant exhaust velocity. Since the ultimate fate of the propellant is not a concern, the fastest possible burn is usually the optimal procedure.
To impart the most kinetic energy to a spacecraft whose free-fall velocity varies with time, the burn must occur when the spacecraft is moving fastest, which usually occurs at periapsis (the point of closest approach).
There are also proposals to use aerodynamic lift at the point of closest approach (an aerogravity assist), to achieve a larger deflection and hence more energy gain.
Read more about this topic: Gravity Assist