Most FM tuners are extremely sensitive and do not need a booster. If you boost a weak signal then the RF noise will also be boosted. A cheap little FM radio might be too simple and be overloaded by other more powerful stations.
I have an old Sony Walkman AM-FM portable radio. It is severely overloaded by strong local stations when it is switched to "DX" (distant). It is not overloaded when switched to "LOCAL" because then the RF input is attenuated but then it cannot pickup weak stations.
With FM the physics is similar like with navigation light tower. Or You see the light and no amplifier is needed for it, or You not see it and then no any even million-fold amplification will not help, because it had been sinked beyond the horizon. So, the only real cure is to erase Your antenna higher, for example lift it from 2 meters to 20 meters widden distance about 4,5-fold. So, dare it! And next thing ought to check is SDR tuner like those 5 dollar blue wonder (I mean RTL28332 or similar, up even to bold priced Hack-RF1), there the amplification is adjustable as well the plethora of different kind of filters are exstant (if only Your computer is mighty enough to re-calculate all those sounds in real time scale).