Lowest voltage drop diode possible

Check out the SM74611 Smart Bypass Diode from Texas Instruments.

Forward Voltage:
Vf[V] = 26mV @ 8A, Tj = 25°C

Other alternatives:

LX2400 Cool bypass switch (CBS) from Microsemi

Typical Forward Voltage
VF = 50mV @ 10A, Tamb = 85°C

SPV1001 Cool bypass switch (CBS) from STMicroelectronics

Vf[V] = 120mV @ 8A, Tj = 25°C
Vf[V] = 270mV @ 8A, Tj = 125°C

SBR30U30CT Super Barrier Rectifier from Diodes

Vf[V] = 190mV @ 2.5A, 125°C
Vf[V] = 250mV @ 5A, 125°C

An ideal diode controller and MOSFET can be applied this situation -- the net effect is that of a Iload*Rds(on) voltage drop diode. Probably the simplest to apply would be Linear's LTC4412.

Dedicated supercapacitor charger ICs also likely would solve the issue, but would require careful specification.

If you add few turns of wire to your antenna coil probably will get higher voltages and lower currents so you could employ Schottky diodes. Impedance matching is very important in harvesting RF energy. Some ferrite core could also help because will capture more energy. Energy required to switch a synchronous Mosfet rectifier at 13 MHz is probably more than the energy harvested.