would the reader read the card again once restarted
Probably.
If so can this be done automatically using code?
I don't know.
maybe switching off and on the 5v pin
That might do it, although it is possible that the reader could wait until a card is out of the field before continuing with it's initialisation. Some readers use the turn on time to set calibration parameters to counter effects of local metal fittings. But a reader could take 1.5 seconds to actually read when first switched on. Their are so many unknowns with the reader.