At what point during an action potential are the sodium potassium pumps working?
The Sodium-Potassium Pumps are always at work. One can think of them as a continuous process that maintains the equilibrium potential for the individual ions. They always are grabbing internal sodium and exchanging it with external potassium at the cost of ATP. 

However a neuron's rest state (in your example -60 mV) is a combination of the equilibrium of the Sodium, Potassium, Chlorine, and other ions. Thus when the membrane hyperpolarizes beyond the rest potential, it is actually the leak potential that brings the membrane potential back up, not the Sodium-Potassium pump. 

Leak potentials arise from ions (usually chorine) that pass through the membrane via channels that are always open. Furthermore, sodium channels reactivate and a small amount open to  sodium to enter. (Recall as a population there is usually a small amount of sodium channels open at rest. Another contributing factor is as the potassium channels close the other to factors dominate and slowly bring the membrane back to rest state. Remember that rest is defined as the balance of currents, there will always be small amounts of currents flowing. So even though the neuron is rest small amounts of sodium leak in and less potassium out. The sodium potassium pump maintains the equilbrium potential that allows these currents to flow.

The Sodium-Potassium pump is a slower process, so it usually can be ignored over a single spike. But If there is a high frequency spike train then the small amount of sodium that enters the cell and potassium that exits the cell can add up and effect the equilibrium potentials of the individual Ions. This of course changes the firing properties of the neuron.