How to filter/clamp/suppress positive voltage spikes from IC
The answer to your question would be "add sufficient decoupling capacitors": As you know, the current that flows into a capacitor is proportional to the time-derivative of the voltage across it. So, if there is a sudden voltage spike, a capacitor is able to absorb that by storing its energy.

You're not telling us anything about the energy of these spikes, but seeing that this is RFID, they probably won't be too bad: place a 100 nF ceramic cap really close to the IC between supply and ground, and complete with a 1 µF (which can slightly further away – avoid vias between capacitor and IC). 

Now, I hear that "capacitance multiplier" method once in a while to tackle insufficiently stable power supplies (i.e. supplies that can't deal with load changes / spikes). To me, that sounds like a "textbook method", introduced by someone who just got access to an early transistor in the 1960s and built something really useful for the time.

A modern LDO (low-dropout regulator) is a device actually built to fulfill exactly the role that your capacitance multiplier is supposed to fulfill: stabilize an output voltage with a limited capacitance. Buy a modern LDO (e.g. an LP4907) for a couple of cent: they have excellent noise rejection properties, and come in tiny packages (assuming your RFID reader doesn't need large currents). You can place them close to he IC, so that the trace inductance doesn't become large.