RFID reader(125kHz) switching between antennas
It always helps to find the datasheet of the parts you use:



That's from the datasheet.  It says there's two types of reader:


Read once, must move card out of range to read again.
Read continuously - card data will be returned as long as it can be read.


You seem to have the first type, which is the standard type.

If you want to read continuosly, you'll need to either get the version of the RDM6300 that reads continuously or you'll need to get a different reader.

That it can't read a card more than once has nothing to do with providing enough power.  It has to do with the reader rejecting tags it has already read.