You might find that your local radio station will blast out far more power than your own transmitter, even though it's in the same house. When I was building crystal radios back in 1965, I could get the local 1000-watt KADY in St. Charles (MO) but not the clear-channel KMOX with its 50KW in St. Louis, just over the river.
Using a battery powered RF amp is unfair as it takes the circuit out of the crystal radio category -- after all, every AM radio has a detector!
One design that keeps it "free power" uses two antennas. One long one that you tune to a strong station, rectify using a voltage doubler or tripler and then filter and use THAT voltage to power the RF or audio amplifier. A second antenna and circuit is used for the regular crystal radio portion to tune the station of desired interest.
Another design biases the diode (again, a battery, but it's being used in a different way) up so that it begins rectifying with smaller signals instead of waiting for something as big as 0.3v to break the barrier potential. Never tried a hot-carrier (Shottky) diode as they weren't around in 1965.
Whatever design you use for your crystal radio, find one with lots of tuned stages. Most simple crystal radios made using single-stage toilet paper tube coils and such don't tune well (or at all) and if you have more than one station, you can't separate them.
Dean